Linear Algebra – Mathematics Optional PYQs (2008–2026)

Here are all 118 Linear Algebra questions asked in UPSC Mathematics Optional Paper I from 2008 to 2026, arranged under the 4 topics of the official syllabus. Within each topic the latest questions come first.

Each question ends with a code such as [2024/10m/1a] – the year, the marks and the question number in the paper.

1.1] Vector spaces

26 questions

1. Given $P_3[x]=\{a_0+a_{1}x+a_2{x}^2+a_3{x}^3\mid a_0,a_1,a_2,a_3\in ℝ\}$ as a vector space over the field $ℝ$. (i) Let $H=\{p\left(x\right)\in P_3[x]\mid {\int }_{-1}^{1}p\left(x\right)dx=0\}$. Prove that $H$ is a subspace of $P_3[x]$ and find a basis of $H$. (ii) Find a subspace $K$ of $P_3[x]$ such that $P_3[x]=H\oplus K$. [2026/15m/3a]

2. Can the set $\{(0,0,0,3),(1,1,0,0),(0,1,-1,0)\}$ be extended to form a basis of the vector space ${ℝ}^4$? Justify your answer. [2025/10m/1a]

3. Let $P_n$ denote the vector space of all polynomials of degree $\le n$ over $ℝ$. Verify that $dim\left(\frac{P_4}{P_2}\right)=dim{P}_4-dim{P}_2.$ [2025/8m/4c(ii)]

4. Let $H$ be a subspace of ${ℝ}^4$ spanned by the vectors $v_1=(1,-2,5,-3)$, $v_2=(2,3,1,-4)$, $v_3=(3,8,-3,-5)$. Then find a basis and dimension of $H$, and extend the basis of $H$ to a basis of ${ℝ}^4$. [2024/10m/1a]

5. Let $V_1=(2,-1,3,2)$, $V_2=(-1,1,1,-3)$ and $V_3=(1,1,9,-5)$ be three vectors of the space ${ℝ}^4$. Does $(3,-1,0,-1)\in span\{V_1,V_2,V_3\}$? Justify your answer. [2023/10m/1a]

6. Prove that any set of $n$ linearly independent vectors in a vector space $V$ of dimension $n$ constitutes a basis for $V$. [2022/10m/1a]

7. Let the set $P=\left\{\left(\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right)\right|\begin{array}{c}\hfill x-y-z=0\mathrm{\text{and}}\hfill \\ \hfill 2x-y+z=0\hfill \end{array}\}$ be the collection of vectors of a vector space ${ℝ}^3\left(ℝ\right)$. Then (i) prove that $P$ is a subspace of ${ℝ}^3$. (ii) find a basis and dimension of $P$. [2022/20m/3a]

8. Show that $S=\{(x,2y,3x):x,y\mathrm{\text{are real numbers}}\}$ is a subspace of ${ℝ}^3\left(ℝ\right)$. Find two bases of $S$. Also find the dimension of $S$. [2021/15m/2c]

9. Consider the set $V$ of all $n\times n$ real magic squares. Show that $V$ is a vector space over $ℝ$. Give examples of two distinct $2\times 2$ magic squares. [2020/10m/1a]

10. Express basis vectors $e_1=(1,0)$ and $e_2=(0,1)$ as linear combinations of ${\alpha }_1=(2,-1)$ and ${\alpha }_2=(1,3)$. [2018/10m/1b]

11. Suppose $U$ and $W$ are distinct four dimensional subspaces of a vector space $V$, where $dimV=6$. Find the possible dimensions of subspace $U\cap W$. [2017/10m/2d]

12. If $\begin{array}{cc}\hfill W_1\hfill & \hfill =\{(x,y,z)\mid x+y-z=0\}\hfill \\ \hfill W_2\hfill & \hfill =\{(x,y,z)\mid 3x+y-2z=0\}\hfill \\ \hfill W_3\hfill & \hfill =\{(x,y,z)\mid x-7y+3z=0\}\hfill \end{array}$ then find $dim(W_1\cap W_2\cap W_3)$ and $dim(W_1+W_2)$. [2016/3m/1b(ii)]

13. The vectors $V_1=(1,1,2,4)$, $V_2=(2,-1,-5,2)$, $V_3=(1,-1,-4,0)$ and $V_4=(2,1,1,6)$ are linearly independent. Is it true? Justify your answer. [2015/10m/1a]

14. Find the dimension of the subspace of ${ℝ}^4$, spanned by the set $\{(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)\}$ Hence find its basis. [2015/12m/4b]

15. Find one vector in ${ℝ}^3$ which generates the intersection of $V$ and $W$, where $V$ is the $xy$ plane and $W$ is the space generated by the vectors $(1,2,3)$ and $(1,-1,1)$. [2014/10m/1a]

16. Let $V$ and $W$ be the following subspaces of ${ℝ}^4$ : $V=\{(a,b,c,d):b-2c+d=0\}$ and $W=\{(a,b,c,d):a=d,b=2c\}.$ Find a basis and the dimension of (i) $V$, (ii) $W$, (iii) $V\cap W$. [2014/15m/2a]

17. Show that the vectors $X_1=(1,1+i,i)$, $X_2=(i,-i,1-i)$ and $X_3=(0,1-2i,2-i)$ in ${ℂ}^3$ are linearly independent over the field of real numbers but are linearly dependent over the field of complex numbers. [2013/8m/2c(ii)]

18. Prove or disprove the following statement : If $B=\{b_1,b_2,b_3,b_4,b_5\}$ is a basis for ${ℝ}^5$ and $V$ is a two-dimensional subspace of ${ℝ}^5$, then $V$ has a basis made of just two members of $B$. [2012/12m/1c]

19. Let $V$ be the vector space of all $2\times 2$ matrices over the field of real numbers. Let $W$ be the set consisting of all matrices with zero determinant. Is $W$ a subspace of $V$? Justify your answer. [2012/8m/2a(i)]

20. Show that the subspaces of ${ℝ}^3$ spanned by two sets of vectors $\{(1,1,-1),(1,0,1)\}$ and $\{(1,2,-3),(5,2,1)\}$ are identical. Also find the dimension of this subspace. [2011/10m/2b(i)]

21. (i) In the $n$-space ${ℝ}^n$, determine whether or not the set $\{e_1-e_2,e_2-e_3,\cdots ,e_{n-1}-e_n,e_n-e_1\}$ is linearly independent. (ii) Let $T$ be a linear transformation from a vector space $V$ over reals into $V$ such that $T-T^2=I$. Show that $T$ is invertible. [2010/20m/4a]

22. Show that the set of solutions of the homogeneous linear differential equation $y\prime +p\left(x\right)y=0$ on an interval $I=[a,b]$ forms a vector subspace $W$ of the real vector space of continuous functions on $I$. What is the dimension of $W$? [2010/20m/7a]

23. Prove that the set $V$ of the vectors $(x_1,x_2,x_3,x_4)$ in ${ℝ}^4$ which satisfy the equations $x_1+x_2+2x_3+x_4=0$ and $2x_1+3x_2-x_3+x_4=0$, is a subspace of ${ℝ}^4$. What is the dimension of this subspace? Find one of its bases. [2009/12m/1b]

24. Prove that the set $V$ of all $3\times 3$ real symmetric matrices forms a linear subspace of the space of all $3\times 3$ real matrices. What is the dimension of this subspace? Find at least one of the bases for $V$. [2009/20m/4a]

25. Let $S$ be a non-empty set and let $V$ denote the set of all functions from $S$ into $ℝ$. Show that $V$ is a vector space with respect to the vector addition $(f+g)\left(x\right)=f\left(x\right)+g\left(x\right)$ and scalar multiplication $\left(c.f\right)\left(x\right)=cf\left(x\right)$. [2008/12m/1b]

26. Find the dimension of the subspace of ${ℝ}^4$ spanned by the set $\{(1,0,0,0),(0,1,0,0),(1,2,0,1),(0,0,0,1)\}$. Hence find a basis for the subspace. [2008/20m/4a]

1.2] Linear transformations

31 questions

1. Let $F$ be a subfield of complex numbers and let $T$ be the linear transformation from $F^3$ into $F^3$, defined by $T(x,y,z)=(x-y+2z,2x+y,-x-2y+2z)$ If $(a,b,c)$ is a vector in $F^3$, then answer the following : (i) Find the condition on $a$, $b$, $c$ such that the vector $(a,b,c)$ be in the null space of $T$. What is the nullity of $T$? (ii) Find the condition on $a$, $b$, $c$ such that the vector $(a,b,c)$ be in the range of $T$. What is the rank of $T$? [2026/15m/4a]

2. Find the range, rank, kernel and nullity of the linear transformation $T:{ℝ}^4\to {ℝ}^3$ given by $T(x,y,z,w)=(x-w,y+z,z-w)$. [2025/10m/1b]

3. Let $T:{ℝ}^3\to {ℝ}^2$ be a linear transformation such that $T(1,1,-1)=(1,0)$, $T(4,1,1)=(0,1)$ and $T(1,-1,2)=(1,1)$. Find $T$. [2025/15m/2a]

4. Let $T:{ℝ}^3\to {ℝ}^3$ be a linear operator and $B=\{v_1,v_2,v_3\}$ be a basis of ${ℝ}^3$ over $ℝ$. Suppose that $T{v}_1=(1,1,0)$, $T{v}_2=(1,0,-1)$, $T{v}_3=(2,1,-1)$. Find a basis for the range space and null space of $T$. [2024/10m/1b]

5. Consider a linear operator $T$ on ${ℝ}^3$ over $ℝ$ defined by $T(x,y,z)=(2x,4x-y,2x+3y-z)$. Is $T$ invertible? If yes, justify your answer and find $T^{-1}$. [2024/15m/2a]

6. Let $V=M_{2\times 2}\left(ℝ\right)$ denote a vector space over the field of real numbers. Find the matrix of the linear mapping $\varphi :V\to V$ given by $\varphi \left(v\right)=\left(\begin{array}{cc}\hfill 1\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill -1\hfill \end{array}\right)v$ with respect to standard basis of $M_{2\times 2}\left(ℝ\right)$, and hence find the rank of $\varphi$. Is $\varphi$ invertible? Justify your answer. [2024/15m/3a]

7. Find the rank and nullity of the linear transformation: $T:{ℝ}^3\to {ℝ}^3$ given by $T(x,y,z)=(x+z,x+y+2z,2x+y+3z)$ [2023/10m/1b]

8. If the matrix of a linear transformation $T:{ℝ}^3\to {ℝ}^3$ relative to the basis $\{(1,0,0),(0,1,0),(0,0,1)\}$ is $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 2\hfill \\ \hfill -1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right],$ then find the matrix of $T$ relative to the basis $\{(1,1,1),(0,1,1),(0,0,1)\}$. [2023/15m/2a]

9. Let $T:{ℝ}^2\to {ℝ}^3$ be a linear transformation such that $T\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 0\hfill \end{array}\right)=\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 3\hfill \end{array}\right)$ and $T\left(\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right)=\left(\begin{array}{c}\hfill -3\hfill \\ \hfill 2\hfill \\ \hfill 8\hfill \end{array}\right)$. Find $T\left(\begin{array}{c}\hfill 2\hfill \\ \hfill 4\hfill \end{array}\right)$. [2022/10m/1b]

10. Find a linear map $T:{ℝ}^2\to {ℝ}^2$ which rotates each vector of ${ℝ}^2$ by an angle $\theta$. Also, prove that for $\theta =\frac{\pi }{2}$, $T$ has no eigenvalue in $ℝ$. [2022/15m/4a]

11. Find the matrix associated with the linear operator on $V_3\left(ℝ\right)$ defined by $T(a,b,c)=(a+b,a-b,2c)$ with respect to the ordered basis $B=\{(0,1,1),(1,0,1),(1,1,0)\}$. [2021/10m/1b]

12. Let $M_2\left(ℝ\right)$ be the vector space of all $2\times 2$ real matrices. Let $B=\left[\begin{array}{cc}\hfill 1\hfill & \hfill -1\hfill \\ \hfill -4\hfill & \hfill 4\hfill \end{array}\right]$. Suppose $T:M_2\left(ℝ\right)\to M_2\left(ℝ\right)$ is a linear transformation defined by $T\left(A\right)=BA$. Find the rank and nullity of $T$. Find a matrix $A$ which maps to the null matrix. [2020/10m/1b]

13. Let $F$ be a subfield of complex numbers and $T$ a function from $F^3\to F^3$ defined by $T(x_1,x_2,x_3)=(x_1+x_2+3x_3,2x_1-x_2,-3x_1+x_2-x_3)$. What are the conditions on $a$, $b$, $c$ such that $(a,b,c)$ be in the null space of $T$? Find the nullity of $T$. [2020/15m/3b]

14. Let $T:{ℝ}^2\to {ℝ}^2$ be a linear map such that $T(2,1)=(5,7)$ and $T(1,2)=(3,3)$. If $A$ is the matrix corresponding to $T$ with respect to the standard bases $e_1$, $e_2$, then find Rank $\left(A\right)$. [2019/10m/1c]

15. Consider the matrix mapping $A:{ℝ}^4\to {ℝ}^3$, where $A=\left(\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill -2\hfill \\ \hfill 3\hfill & \hfill 8\hfill & \hfill 13\hfill & \hfill -3\hfill \end{array}\right)$. Find a basis and dimension of the image of $A$ and those of the kernel $A$. [2017/15m/3a]

16. If $M_2\left(ℝ\right)$ is space of real matrices of order $2\times 2$ and $P_2\left(x\right)$ is the space of real polynomials of degree at most 2, then find the matrix representation of $T:M_2\left(ℝ\right)\to P_2\left(x\right)$, such that $T\left(\left[\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right]\right)=a+c+(a-d)x+(b+c)x^2$, with respect to the standard bases of $M_2\left(ℝ\right)$ and $P_2\left(x\right)$. Further find the null space of $T$. [2016/10m/2a(i)]

17. If $T:P_2\left(x\right)\to P_3\left(x\right)$ is such that $T\left(f\left(x\right)\right)=f\left(x\right)+5{\int }_0^{x}f\left(t\right)dt$, then choosing $\{1,1+x,1-x^2\}$ and $\{1,x,x^2,x^3\}$ as bases of $P_2\left(x\right)$ and $P_3\left(x\right)$ respectively, find the matrix of $T$. [2016/6m/2a(ii)]

18. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 2\hfill \\ \hfill -2\hfill & \hfill 1\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}\right]$ is the matrix representation of a linear transformation $T:P_2\left(x\right)\to P_2\left(x\right)$ with respect to the bases $\{1-x,x(1-x),x(1+x)\}$ and $\{1,1+x,1+x^2\}$, then find $T$. [2016/18m/2c]

19. Let $V={ℝ}^3$ and $T\in A\left(V\right)$, for all $a_i\in A\left(V\right)$, be defined by $T(a_1,a_2,a_3)=(2a_1+5a_2+a_3,-3a_1+a_2-a_3,-a_1+2a_2+3a_3)$ What is the matrix $T$ relative to the basis $V_1=(1,0,1)V_2=(-1,2,1)V_3=(3,-1,1)?$ [2015/12m/3a]

20. Let $P_n$ denote the vector space of all real polynomials of degree atmost $n$ and $T:P_2\to P_3$ be a linear transformation given by $T\left(p\left(x\right)\right)={\int }_0^{x}p\left(t\right)dt,p\left(x\right)\in P_2.$ Find the matrix of $T$ with respect to the bases $\{1,x,x^2\}$ and $\{1,x,1+x^2,1+x^3\}$ of $P_2$ and $P_3$ respectively. Also, find the null space of $T$. [2013/10m/2a(i)]

21. Let $V$ be an $n$-dimensional vector space and $T:V\to V$ be an invertible linear operator. If $\beta =\{X_1,X_2,\dots ,X_n\}$ is a basis of $V$, show that $\beta \prime =\{T{X}_1,T{X}_2,\dots ,T{X}_n\}$ is also a basis of $V$. [2013/8m/2a(ii)]

22. Let $T:{ℝ}^3\to {ℝ}^3$ be the linear transformation defined by $T(\alpha ,\beta ,\gamma )=(\alpha +2\beta -3\gamma ,2\alpha +5\beta -4\gamma ,\alpha +4\beta +\gamma )$ Find a basis and the dimension of the image of $T$ and the kernel of $T$. [2012/12m/1d]

23. Consider the linear mapping $f:{ℝ}^2\to {ℝ}^2$ by $f(x,y)=(3x+4y,2x-5y)$ Find the matrix $A$ relative to the basis $\{(1,0),(0,1)\}$ and the matrix $B$ relative to the basis $\{(1,2),(2,3)\}$. [2012/12m/2b(i)]

24. Find the nullity and a basis of the null space of the linear transformation $A:{ℝ}^{\left(4\right)}\to {ℝ}^{\left(4\right)}$ given by the matrix $A=\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \end{array}\right].$ [2011/10m/2b(ii)]

25. Show that the vectors $(1,1,1)$, $(2,1,2)$ and $(1,2,3)$ are linearly independent in ${ℝ}^{\left(3\right)}$. Let $T:{ℝ}^{\left(3\right)}\to {ℝ}^{\left(3\right)}$ be a linear transformation defined by $T(x,y,z)=(x+2y+3z,x+2y+5z,2x+4y+6z).$ Show that the images of above vectors under $T$ are linearly dependent. Give the reason for the same. [2011/10m/2c(i)]

26. What is the null space of the differentiation transformation $\frac{d}{dx}:P_n\to P_n$ where $P_n$ is the space of all polynomials of degree $\le n$ over the real numbers? What is the null space of the second derivative as a transformation of $P_n$? What is the null space of the $k$th derivative? [2010/12m/1b]

27. Let $M=\left(\begin{array}{ccc}\hfill 4\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 3\hfill \end{array}\right)$. Find the unique linear transformation $T:{ℝ}^3\to {ℝ}^3$ so that $M$ is the matrix of $T$ with respect to the basis $\beta =\{v_1=(1,0,0),v_2=(1,1,0),v_3=(1,1,1)\}$ of ${ℝ}^3$ and $\beta \prime =\{w_1=(1,0),w_2=(1,1)\}$ of ${ℝ}^2$. Also find $T(x,y,z)$. [2010/20m/2a]

28. (i) In the $n$-space ${ℝ}^n$, determine whether or not the set $\{e_1-e_2,e_2-e_3,\cdots ,e_{n-1}-e_n,e_n-e_1\}$ is linearly independent. (ii) Let $T$ be a linear transformation from a vector space $V$ over reals into $V$ such that $T-T^2=I$. Show that $T$ is invertible. [2010/20m/4a]

29. Let $ℬ=\{(1,1,0),(1,0,1),(0,1,1)\}$ and $ℬ\prime =\{(2,1,1),(1,2,1),(-1,1,1)\}$ be the two ordered bases of ${ℝ}^3$. Then find a matrix representing the linear transformation $T:{ℝ}^3\to {ℝ}^3$ which transforms $ℬ$ into $ℬ\prime$. Use this matrix representation to find $T\left(\bar{x}\right)$, where $\bar{x}=(2,3,1)$. [2009/20m/2a]

30. Let $L:{ℝ}^4\to {ℝ}^3$ be a linear transformation defined by $L\left((x_1,x_2,x_3,x_4)\right)=(x_3+x_4-x_1-x_2,x_3-x_2,x_4-x_1)$ Then find the rank and nullity of $L$. Also, determine null space and range space of $L$. [2009/20m/3a]

31. Show that $B=\{(1,0,0),(1,1,0),(1,1,1)\}$ is a basis of ${ℝ}^3$. Let $T:{ℝ}^3\to {ℝ}^3$ be a linear transformation such that $T(1,0,0)=(1,0,0)$, $T(1,1,0)=(1,1,1)$ and $T(1,1,1)=(1,1,0)$. Find $T(x,y,z)$. [2008/20m/2a]

1.3] Algebra of matrices and linear equations

26 questions

1. Find all the values of $a$ for which the system of homogeneous equations $\begin{array}{cc}\hfill ax+y+z\hfill & \hfill =0\hfill \\ \hfill x+y-z\hfill & \hfill =0\hfill \\ \hfill x+y-az\hfill & \hfill =0\hfill \end{array}$ has non-trivial solutions, and hence determine all the solutions for each value of $a$. [2026/10m/1a]

2. Reduce the following matrix to echelon form : $A=\left[\begin{array}{cccc}\hfill 2\hfill & \hfill -2\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill -3\hfill & \hfill 6\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill -7\hfill & \hfill 10\hfill & \hfill 2\hfill \end{array}\right]$ [2025/15m/3a]

3. Find the rank of the matrix $A=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill -1\hfill & \hfill 3\hfill & \hfill 0\hfill & \hfill -4\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 3\hfill & \hfill -2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$ by reducing it to row-reduced echelon form. [2023/15m/4a]

4. Find all solutions to the following system of equations by row-reduced method : $\begin{array}{cc}\hfill x_1+2x_2-x_3\hfill & \hfill =2\hfill \\ \hfill 2x_1+3x_2+5x_3\hfill & \hfill =5\hfill \\ \hfill -x_1-3x_2+8x_3\hfill & \hfill =-1\hfill \end{array}$ [2022/15m/2a]

5. For two square matrices $A$ and $B$ of order 2, show that trace $\left(AB\right)=$ trace $\left(BA\right)$. Hence show that $AB-BA\ne I_2$, where $I_2$ is an identity matrix of order 2. [2021/7m/3c(ii)]

6. Reduce the following matrix to a row-reduced echelon form and hence also, find its rank : $A=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 2\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 6\hfill & \hfill 2\hfill & \hfill 6\hfill & \hfill 2\hfill \\ \hfill 3\hfill & \hfill 9\hfill & \hfill 1\hfill & \hfill 10\hfill & \hfill 6\hfill \end{array}\right]$ [2021/10m/4a(i)]

7. Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 3\hfill \\ \hfill 4\hfill & \hfill 1\hfill & \hfill 8\hfill \end{array}\right]\mathrm{\text{and}}B=\left[\begin{array}{ccc}\hfill -11\hfill & \hfill 2\hfill & \hfill 2\hfill \\ \hfill -4\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 6\hfill & \hfill -1\hfill & \hfill -1\hfill \end{array}\right].$ \(i\) Find $AB$. (ii) Find $det\left(A\right)$ and $det\left(B\right)$. (iii) Solve the following system of linear equations : $x+2z=3,2x-y+3z=3,4x+y+8z=14$ [2020/15m/4a]

8. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -4\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 0\hfill & \hfill -3\hfill \end{array}\right]\mathrm{\text{and}}B=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill -1\hfill \end{array}\right]$ then show that $AB=6I_3$. Use this result to solve the following system of equations : $\begin{array}{cc}\hfill 2x+y+z\hfill & \hfill =5\hfill \\ \hfill x-y\hfill & \hfill =0\hfill \\ \hfill 2x+y-z\hfill & \hfill =1\hfill \end{array}$ [2019/10m/1d]

9. Let $A=\left(\begin{array}{cccc}\hfill 5\hfill & \hfill 7\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -8\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill -3\hfill & \hfill 1\hfill \end{array}\right)$ \(i\) Find the rank of matrix $A$. (ii) Find the dimension of the subspace $V=\{(x_1,x_2,x_3,x_4)\in {ℝ}^4|A\left(\begin{array}{c}\hfill x_1\hfill \\ \hfill x_2\hfill \\ \hfill x_3\hfill \\ \hfill x_4\hfill \end{array}\right)=0\}$ [2019/20m/3c]

10. Let $A$ be a $3\times 2$ matrix and $B$ a $2\times 3$ matrix. Show that $C=A\cdot B$ is a singular matrix. [2018/10m/1a]

11. For the system of linear equations $\begin{array}{cc}\hfill x+3y-2z\hfill & \hfill =-1\hfill \\ \hfill 5y+3z\hfill & \hfill =-8\hfill \\ \hfill x-2y-5z\hfill & \hfill =7\hfill \end{array}$ determine which of the following statements are true and which are false : (i) The system has no solution. (ii) The system has a unique solution. (iii) The system has infinitely many solutions. [2018/13m/3a]

12. Consider the following system of equations in $x$, $y$, $z$ : $\begin{array}{cc}\hfill x+2y+2z\hfill & \hfill =1\hfill \\ \hfill x+ay+3z\hfill & \hfill =3\hfill \\ \hfill x+11y+az\hfill & \hfill =b.\hfill \end{array}$ \(i\) For which values of $a$ does the system have a unique solution? (ii) For which pair of values $(a,b)$ does the system have more than one solution? [2017/15m/4b]

13. Using elementary row operations, find the inverse of $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$. [2016/6m/1a(i)]

14. Using elementary row operations, find the condition that the linear equations $\begin{array}{cc}\hfill x-2y+z\hfill & \hfill =a\hfill \\ \hfill 2x+7y-3z\hfill & \hfill =b\hfill \\ \hfill 3x+5y-2z\hfill & \hfill =c\hfill \end{array}$ have a solution. [2016/7m/1b(i)]

15. Reduce the following matrix to row echelon form and hence find its rank : $\left[\begin{array}{cccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 1\hfill & \hfill 5\hfill & \hfill 5\hfill & \hfill 7\hfill \\ \hfill 8\hfill & \hfill 1\hfill & \hfill 14\hfill & \hfill 17\hfill \end{array}\right].$ [2015/10m/1b]

16. Using elementary row or column operations, find the rank of the matrix $\left[\begin{array}{cccc}\hfill 0\hfill & \hfill 1\hfill & \hfill -3\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -2\hfill & \hfill 0\hfill \end{array}\right].$ [2014/10m/1b]

17. Investigate the values of $\lambda$ and $\mu$ so that the equations $x+y+z=6$, $x+2y+3z=10$, $x+2y+\lambda z=\mu$ have (1) no solution, (2) a unique solution, (3) an infinite number of solutions. [2014/10m/2b(i)]

18. Find the inverse of the matrix : $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 3\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 7\hfill \\ \hfill 3\hfill & \hfill 2\hfill & \hfill -1\hfill \end{array}\right]$ by using elementary row operations. Hence solve the system of linear equations $\begin{array}{cc}\hfill x+3y+z\hfill & \hfill =10\hfill \\ \hfill 2x-y+7z\hfill & \hfill =21\hfill \\ \hfill 3x+2y-z\hfill & \hfill =4\hfill \end{array}$ [2013/10m/1a]

19. Find the rank of the matrix $A=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 2\hfill & \hfill 3\hfill & \hfill 5\hfill & \hfill 8\hfill & \hfill 12\hfill \\ \hfill 3\hfill & \hfill 5\hfill & \hfill 8\hfill & \hfill 12\hfill & \hfill 17\hfill \\ \hfill 5\hfill & \hfill 8\hfill & \hfill 12\hfill & \hfill 17\hfill & \hfill 23\hfill \\ \hfill 8\hfill & \hfill 12\hfill & \hfill 17\hfill & \hfill 23\hfill & \hfill 30\hfill \end{array}\right]$ [2013/8m/2b(ii)]

20. Find the dimension and a basis for the space $W$ of all solutions of the following homogeneous system using matrix notation : $\begin{array}{cc}\hfill x_1+2x_2+3x_3-2x_4+4x_5\hfill & \hfill =0\hfill \\ \hfill 2x_1+4x_2+8x_3+x_4+9x_5\hfill & \hfill =0\hfill \\ \hfill 3x_1+6x_2+13x_3+4x_4+14x_5\hfill & \hfill =0\hfill \end{array}$ [2012/12m/2a(ii)]

21. Let $H=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill i\hfill & \hfill 2+i\hfill \\ \hfill -i\hfill & \hfill 2\hfill & \hfill 1-i\hfill \\ \hfill 2-i\hfill & \hfill 1+i\hfill & \hfill 2\hfill \end{array}\right)$ be a Hermitian matrix. Find a non-singular matrix $P$ such that $D=P^{T}H\bar{P}$ is diagonal. [2012/20m/2c]

22. Let $A$ be a non-singular, $n\times n$ square matrix. Show that $A\cdot \left(adjA\right)=\left|A\right|\cdot I_n$. Hence show that $\left|adj\left(adjA\right)\right|={\left|A\right|}^{{(n-1)}^2}$. [2011/10m/1a]

23. Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill \\ \hfill 0\hfill & \hfill 6\hfill & \hfill 7\hfill \end{array}\right],X=\left[\begin{array}{c}\hfill x\hfill \\ \hfill y\hfill \\ \hfill z\hfill \end{array}\right],B=\left[\begin{array}{c}\hfill 2\hfill \\ \hfill 6\hfill \\ \hfill 5\hfill \end{array}\right].$ Solve the system of equations given by $AX=B$ Using the above, also solve the system of equations $A^{T}X=B$ where $A^T$ denotes the transpose of matrix $A$. [2011/10m/1b]

24. Let $A$ and $B$ be $n\times n$ matrices over reals. Show that $I-BA$ is invertible if $I-AB$ is invertible. Deduce that $AB$ and $BA$ have the same eigenvalues. [2010/20m/3a]

25. Show that the matrix $A$ is invertible if and only if the $adj\left(A\right)$ is invertible. Hence find $\left|adj\left(A\right)\right|$. [2008/12m/1a]

26. Let $a$ be a non-singular matrix. Show that if $I+A+A^2+\cdots +A^n=0,$ then $A^{-1}=A^n$. [2008/20m/3a]

1.4] Eigenvalues and special matrices

35 questions

1. If $A$ and $B$ are similar matrices, then show that $A$ and $B$ have the same rank, trace, characteristic polynomial and eigenvalues. [2026/10m/1b]

2. Let $T:P_2[x]\to P_2[x]$ be a linear transformation defined by $T(a_0+a_{1}x+a_2{x}^2)=-(a_0+2a_1+a_2)+(2a_0+3a_1)x-2(a_0+a_1)x^2$ where $P_2[x]$ denotes the set of all polynomials in $x$ of degree $\le 2$ and $a_0,a_1,a_2\in ℝ$. Find the eigenvalues and corresponding eigenvectors of $T$ using the matrix representation of $T$ with respect to the standard basis of $P_2[x]$. [2026/20m/2a]

3. Find the eigenvalues and the corresponding eigenvectors of the matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 2\hfill & \hfill 0\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill -6\hfill \\ \hfill 2\hfill & \hfill -2\hfill & \hfill 3\hfill \end{array}\right].$ [2025/12m/4c(i)]

4. Let $A=\left(\begin{array}{ccc}\hfill 3\hfill & \hfill 2\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 0\hfill & \hfill 2\hfill \\ \hfill 4\hfill & \hfill 2\hfill & \hfill 3\hfill \end{array}\right)$ be a $3\times 3$ matrix. Find the eigenvalues and the corresponding eigenvectors of $A$. Hence find the eigenvalues and the corresponding eigenvectors of $A^{-15}$, where $A^{-15}={\left(A^{-1}\right)}^{15}$. [2024/20m/4a]

5. Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right].$ \(i\) Verify the Cayley-Hamilton theorem for the matrix $A$. (ii) Show that $A^n=A^{n-2}+A^2-I$ for $n\ge 3$, where $I$ is the identity matrix of order 3. Hence, find $A^{40}$. [2023/20m/3a]

6. Find a linear map $T:{ℝ}^2\to {ℝ}^2$ which rotates each vector of ${ℝ}^2$ by an angle $\theta$. Also, prove that for $\theta =\frac{\pi }{2}$, $T$ has no eigenvalue in $ℝ$. [2022/15m/4a]

7. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill \\ \hfill 2\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$ then show that $A^2=A^{-1}$ (without finding $A^{-1}$). [2021/10m/1a]

8. Prove that the eigen vectors, corresponding to two distinct eigen values of a real symmetric matrix, are orthogonal. [2021/8m/3c(i)]

9. Find the eigen values and the corresponding eigen vectors of the matrix $A=\left(\begin{array}{cc}\hfill 0\hfill & \hfill -i\hfill \\ \hfill i\hfill & \hfill 0\hfill \end{array}\right)$, over the complex-number field. [2021/10m/4a(ii)]

10. Define an $n\times n$ matrix as $A=I-2u\cdot u^T$, where $u$ is a unit column vector. (i) Examine if $A$ is symmetric. (ii) Examine if $A$ is orthogonal. (iii) Show that trace $\left(A\right)=n-2$. (iv) Find $A_{3\times 3}$, when $u=\left[\begin{array}{c}\hfill \frac{1}{3}\hfill \\ \hfill \frac{2}{3}\hfill \\ \hfill \frac{2}{3}\hfill \end{array}\right]$. [2020/20m/2b]

11. Let $A$ and $B$ be two orthogonal matrices of same order and $detA+detB=0$. Show that $A+B$ is a singular matrix. [2019/15m/2b]

12. State the Cayley-Hamilton theorem. Use this theorem to find $A^{100}$, where $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ [2019/15m/4a]

13. Show that if $A$ and $B$ are similar $n\times n$ matrices, then they have the same eigenvalues. [2018/12m/2a]

14. Let $A=\left(\begin{array}{cc}\hfill 2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 3\hfill \end{array}\right)$. Find a non-singular matrix $P$ such that $P^{-1}AP$ is a diagonal matrix. [2017/10m/1a]

15. Show that similar matrices have the same characteristic polynomial. [2017/10m/1b]

16. Prove that distinct non-zero eigenvectors of a matrix are linearly independent. [2017/10m/3b]

17. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 5\hfill & \hfill 2\hfill & \hfill 6\hfill \\ \hfill -2\hfill & \hfill -1\hfill & \hfill -3\hfill \end{array}\right]$, then find $A^{14}+3A-2I$. [2016/4m/1a(ii)]

18. If $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]$, then find the eigenvalues and eigenvectors of $A$. [2016/8m/2b(i)]

19. Prove that eigenvalues of a Hermitian matrix are all real. [2016/8m/2b(ii)]

20. If matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \end{array}\right]$ then find $A^{30}$. [2015/12m/2a]

21. Find the eigen values and eigen vectors of the matrix : $\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 3\hfill \\ \hfill 1\hfill & \hfill 5\hfill & \hfill 1\hfill \\ \hfill 3\hfill & \hfill 1\hfill & \hfill 1\hfill \end{array}\right].$ [2015/12m/2c]

22. Verify Cayley — Hamilton theorem for the matrix $A=\left[\begin{array}{cc}\hfill 1\hfill & \hfill 4\hfill \\ \hfill 2\hfill & \hfill 3\hfill \end{array}\right]$ and hence find its inverse. Also, find the matrix represented by $A^5-4A^4-7A^3+11A^2-A-10I.$ [2014/10m/2b(ii)]

23. Let $A=\left[\begin{array}{ccc}\hfill -2\hfill & \hfill 2\hfill & \hfill -3\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill -6\hfill \\ \hfill -1\hfill & \hfill -2\hfill & \hfill 0\hfill \end{array}\right]$. Find the eigen values of $A$ and the corresponding eigen vectors. [2014/8m/3c(i)]

24. Prove that the eigen values of a unitary matrix have absolute value 1. [2014/7m/3c(ii)]

25. Let $A$ be a square matrix and $A^{*}$ be its adjoint, show that the eigenvalues of matrices $A{A}^{*}$ and $A^{*}A$ are real. Further show that trace $\left(A{A}^{*}\right)$ = trace $\left(A^{*}A\right)$. [2013/10m/1b]

26. Let $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill {\omega }^2\hfill & \hfill \omega \hfill \\ \hfill 1\hfill & \hfill \omega \hfill & \hfill {\omega }^2\hfill \end{array}\right]$ where $\omega (\ne 1)$ is a cube root of unity. If ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ denote the eigenvalues of $A^2$, show that $\left|{\lambda }_1\right|+\left|{\lambda }_2\right|+\left|{\lambda }_3\right|\le 9$. [2013/8m/2b(i)]

27. Let $A$ be a Hermetian matrix having all distinct eigenvalues ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$. If $X_1,X_2,\dots ,X_n$ are corresponding eigenvectors then show that the $n\times n$ matrix $C$ whose $k^{\mathrm{\text{th}}}$ column consists of the vector $X_k$ is non singular. [2013/8m/2c(i)]

28. If $\lambda$ is a characteristic root of a non-singular matrix $A$, then prove that $\frac{\left|A\right|}{\lambda }$ is a characteristic root of $AdjA$. [2012/8m/2b(ii)]

29. Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be the eigen values of a $n\times n$ square matrix $A$ with corresponding eigen vectors $X_1,X_2,\dots ,X_n$. If $B$ is a matrix similar to $A$ show that the eigen values of $B$ are same as that of $A$. Also find the relation between the eigen vectors of $B$ and eigen vectors of $A$. [2011/10m/2a(i)]

30. Verify the Cayley-Hamilton theorem for the matrix $A=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 2\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 3\hfill & \hfill -5\hfill & \hfill 1\hfill \end{array}\right].$ Using this, show that $A$ is non-singular and find $A^{-1}$. [2011/10m/2a(ii)]

31. Let $A=\left[\begin{array}{ccc}\hfill 2\hfill & \hfill -2\hfill & \hfill 2\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill 3\hfill & \hfill -1\hfill \end{array}\right]$ and $C$ be a non-singular matrix of order $3\times 3$. Find the eigen values of the matrix $B^3$ where $B=C^{-1}AC$. [2011/10m/2c(ii)]

32. If ${\lambda }_1$, ${\lambda }_2$, ${\lambda }_3$ are the eigenvalues of the matrix $A=\left(\begin{array}{ccc}\hfill 26\hfill & \hfill -2\hfill & \hfill 2\hfill \\ \hfill 2\hfill & \hfill 21\hfill & \hfill 4\hfill \\ \hfill 4\hfill & \hfill 2\hfill & \hfill 28\hfill \end{array}\right)$ show that $\sqrt{{\lambda }_1^2+{\lambda }_2^2+{\lambda }_3^2}\le \sqrt{1949}$ [2010/12m/1a]

33. Let $A$ and $B$ be $n\times n$ matrices over reals. Show that $I-BA$ is invertible if $I-AB$ is invertible. Deduce that $AB$ and $BA$ have the same eigenvalues. [2010/20m/3a]

34. Find a Hermitian and a skew-Hermitian matrix each whose sum is the matrix $\left[\begin{array}{ccc}\hfill 2i\hfill & \hfill 3\hfill & \hfill -1\hfill \\ \hfill 1\hfill & \hfill 2+3i\hfill & \hfill 2\hfill \\ \hfill -i+1\hfill & \hfill 4\hfill & \hfill 5i\hfill \end{array}\right]$ [2009/12m/1a]

35. Find a $2\times 2$ real matrix $A$ which is both orthogonal and skew-symmetric. Can there exist a $3\times 3$ real matrix which is both orthogonal and skew-symmetric? Justify your answer. [2009/20m/2c]

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