GATE Overflow for GATE ESE - Recent questions in Linear Algebra

GATE ES 2026 | Question: 1

Mon, 23 Feb 2026 13:15:48 +0000

If $2 X+\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]=\left[\begin{array}{ll}3 & 8 \\ 7 & 2\end{array}\right]$, then $X$ is $\_\_\_\_$.

$\left[\begin{array}{ll}2 & 5 \\ 5 & 3\end{array}\right]$
$\left[\begin{array}{cc}-1 & 3 \\ 2 & -1\end{array}\right]$
$\left[\begin{array}{cc}1 & 3 \\ 2 & -1\end{array}\right]$
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

GATE ES 2026 | Question: 27

Mon, 23 Feb 2026 13:14:58 +0000

Consider the matrix $A=\left[\begin{array}{ccc}2 & 2 & 3 \\ 2 & 5 & 6 \\ 3 & 4 & 10\end{array}\right]$.

Which ONE of the following options CORRECTLY lists the eigenvalues of $A$?

$1 ; 8-\sqrt{37} ;-8+\sqrt{37}$
$2 ; 8-\sqrt{37} ;-8+\sqrt{37}$
$1 ; 8+\sqrt{37} ; \quad 8-\sqrt{37}$
$2 ; 8+\sqrt{37} ; \quad 8+\sqrt{37}$

GATE ES 2026 | Question: 32

Mon, 23 Feb 2026 13:14:47 +0000

A system of linear equations is given as $\mathrm{A X}=\mathrm{B}$, where $\mathrm{A}=\left[\begin{array}{cc}2 & 5 \\ 5 & -2\end{array}\right], \quad \mathrm{B}=\left[\begin{array}{l}7 \\ 3\end{array}\right]$, and $\mathrm{X}$ is the solution vector.

Which of the following statements is/are TRUE for this system?

Matrix $\mathrm{A}$ is skew-symmetric.
Eigenvalues of matrix $\mathrm{A}$ are real.
The system is homogeneous.
The system is singular.

GATE ES 2025 | Question: 38

Thu, 06 Mar 2025 16:21:14 +0000

The following is a system of linear equations
$$\begin{array}{c} x-2 y+z=34 \\ 2 x+y+z=102 \\x+y-3 z=17\end{array}$$

The value of $(x+y+z)$ is $\_\_\_\_\_\_$ (rounded off to two decimal places)

GATE ES 2024 | Question: 15

Mon, 19 Feb 2024 10:24:11 +0000

If $\mathbf{X}$ is a vector, and $\mathbf{A}$ and $\mathbf{B}$ are linear operators; then the correct mathematical relationship(s) is/are

$(\mathbf{A}+\mathbf{B}) \mathbf{X}=\mathbf{A X}+\mathbf{B X}$
$(\lambda \mathbf{A}) \mathbf{X}=\lambda(\mathbf{A X})$
$(\mathbf{A B}) \mathbf{X}=\mathbf{A}(\mathbf{B X})$
$(\mathbf{A}+\mathbf{B}) \mathbf{X}=\mathbf{A}^{\mathrm{T}}\mathbf{X}+\mathbf{B}^{\mathrm{T}} \mathbf{X}$

GATE ES 2024 | Question: 14

Mon, 19 Feb 2024 10:24:11 +0000

The eigen values of the matrix $\left[\begin{array}{ll}4 & 3 \\ 3 & 4\end{array}\right]$ are

GATE ES 2023 | Question: 2

Tue, 23 May 2023 03:12:49 +0000

$\mathbf{P}$ and $\mathbf{Q}$ are square matrices. Consider the following

$\mathrm{X}:\left(\mathbf{P}^{-\mathbf{1}}\right)^{-1}=\mathbf{P}$

Y: Symmetric if $\mathbf{Q}=-\mathbf{Q}^{\mathrm{T}}$

The correct choice is

$\mathrm{X}$ is TRUE; $\mathrm{Y}$ is FALSE
$\mathrm{X}$ is FALSE; $\mathrm{Y}$ is TRUE
Both $\mathrm{X}$ and $\mathrm{Y}$ are TRUE
Both $\mathrm{X}$ and $\mathrm{Y}$ are FALSE

GATE ES 2023 | Question: 38

Tue, 23 May 2023 03:12:01 +0000

Consider two matrices $\mathbf{P}=\left[\begin{array}{ll}2 & 3 \\ 1 & 4\end{array}\right]$ and $\mathbf{Q}=\left[\begin{array}{ll}5 & 4 \\ 0 & 2\end{array}\right]$. If $\mathbf{R}=(\mathbf{P Q})^{\mathrm{T}}$ then $\operatorname{det} \mathbf{R}$ is (in integer).

GATE ES 2021 | Question: 26

Tue, 14 Mar 2023 03:44:24 +0000

The $2 \times 2$ matrices $P$ and $Q$ satisfy the following relations:
$\mathbf{P}+\mathbf{Q}=\left[\begin{array}{cc}3 & 1 \\ 2 & 12\end{array}\right]$ and
$\mathbf{P}-\mathbf{Q}=\left[\begin{array}{cc}-1 & -7\\ 8 & 2\end{array}\right]$

The matrix $Q$ is equal to

$\left[\begin{array}{II}2 & 4 \\ -3 & 5\end{array}\right]$
$\left[\begin{array}{II}1 & -3 \\ 5 & 7\end{array}\right]$
$\left[\begin{array}{II}2 & -3 \\ 4 & 5\end{array}\right]$
$\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right]$

GATE ES 2022 | Question: 4

Fri, 17 Feb 2023 07:06:14 +0000

Given $\mathbf{P}$ is $m \times n$ matrix, $\mathbf{Q}$ is $n \times l$ matrix, $\mathbf{R}$ and $\mathbf{S}$ are $n \times n$ matrices, and superscript $\mathrm{T}$ denotes transpose. Consider

Relationship $1$: $(\mathbf{P Q})^{\mathrm{T}}=\mathbf{Q}^{\mathrm{T}} \mathbf{P}^{\mathrm{T}}$

Relationship $2$: $(\mathbf{R S})^{\mathrm{-1}}=\mathbf{S}^{\mathrm{-1}} \mathbf{R}^{\mathrm{-1}}$

Which one of the following is correct?

Relationship $1$ is false; Relationship $2$ is false
Relationship $1$ is true; Relationship $2$ is false
Relationship $1$ is true; Relationship $2$ is true
Relationship $1$ is false; Relationship $2$ is true

GATE ES 2022 | Question: 27

Fri, 17 Feb 2023 07:05:51 +0000

Eigen values of the matrix $\left[\begin{array}{cc}0 & 1 \\ -2 & 3\end{array}\right]$ are

$1$ and $2$
$1$ and $3$
$1$ and $-2$
$0$ and $3$