GATE Overflow for GATE XE - Recent questions in Linear Algebra

GATE XE 2026 | Question: 5

Tue, 24 Feb 2026 15:50:20 +0000

Consider the matrix $A=\left[\begin{array}{ccc}1 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$. Then, which of the following statements is/are true?

All the eigenvalues of $A$ are real numbers
The eigenvalues of $A^{-1}$ are $1, \dfrac{1}{2},-1$
The determinant of $A$ is 3
The matrix $A-2 I$ is invertible, where $I$ denotes the identity matrix of order $3 \times 3$

GATE XE 2026 | Question: 10

Tue, 24 Feb 2026 15:49:48 +0000

Let $\left[\begin{array}{ccc}l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -10\end{array}\right]\left[\begin{array}{ccc}1 & u_{12} & u_{13} \\ 0 & 1 & u_{23} \\ 0 & 0 & 1\end{array}\right]$ be the $L U$ decomposition of the matrix $A=\left[\begin{array}{rrr}1 & 1 & 1 \\ 4 & 3 & -1 \\ 3 & 5 & a\end{array}\right]$. Then, the value of $a$ (in integer) is $\_\_\_\_$

GATE XE 2025 | Question: 14

Sun, 04 May 2025 19:08:39 +0000

Suppose the polynomial $a+b x+c x^{2}+d x^{3}$ interpolates the data, $(-1,1),(0,3),(1,2)$ and $(2,4)$.

Then which one of the following statements is correct?

$a=-2 c, d=-2 b$
$a=2 c, d=2 b$
$b=3 c, a=2 d$
$b=2 c, a=3 d$

GATE XE 2025 | Question: 17

Sun, 04 May 2025 19:08:34 +0000

Suppose that $2$ is an eigenvalue of the matrix

$$\left[\begin{array}{ccc}
0 & 3 & -\alpha \\
0 & 1 & 0 \\
1 & -1 & 3
\end{array}\right]$$

Then the value of $\alpha$ is equal to (Answer in integer) $\_\_\_\_\_$

GATE XE 2025 | Question: 20

Sun, 04 May 2025 19:08:29 +0000

For $a, b \in \mathbb{R}$, consider the system of linear equations

$$ \begin{array}{r} x+y+a z=2 \\ 2 y+2 z=1 \\ a x+2 z=b \end{array}$$

If the system has infinitely many solutions, then which of the following statements is/are correct ?

$a=2, b=3$
$a=2, b=5$
$a=-1, b=-\frac{3}{2}$
$a=3, b=5$

GATE XE 2024 | Question: 5

Sun, 21 Jul 2024 16:42:41 +0000

Let $A$ be a $3 \times 3$ matrix whose eigenvalues are $2,3,4$ and let $I$ be the identity matrix of order 3 . If
\[
A^{-1}=\frac{1}{2 k}\left(A^{2}-9 A\right)+\frac{13}{k} I
\]
for some integer $k \neq 0$, then the value of $k$ is ___________.

GATE XE 2024 | Question: 9

Sun, 21 Jul 2024 16:42:38 +0000

Let $P$ and $Q$ be two square matrices of the same order. Then, which of the following matrices is/are necessarily equal to $(P+2 Q)^{2}$ ?

$P^{2}+4 P Q+4 Q^{2}$
$P(P+2 Q)+Q(2 P+4 Q)$
$(P+2 Q)(2 Q+P)$
$P^{2}+2 P Q+2 Q P+4 Q^{2}$

GATE XE 2023 | Question: 1

Wed, 14 Feb 2024 18:10:17 +0000

Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,$ and $3$. If $B=A^{2}+2 A+I$, where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ are

$4,9,16$
$1,2,3$
$1,4,9$
$4,16,25$

GATE XE 2023 | Question: 4

Wed, 14 Feb 2024 18:10:14 +0000

The second smallest eigenvalue of the eigenvalue problem

$\frac{d^{2} y}{d x^{2}}+(\lambda-3) y=0, \quad y(0)=y(\pi)=0$,

GATE XE 2023 | Question: 8

Wed, 14 Feb 2024 18:10:11 +0000

Let

$P=\left[\begin{array}{lll}4 & -2 & 2 \\ 6 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$, and $Q=\left[\begin{array}{lll}3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5\end{array}\right]$

The eigenvalues of both $P$ and $Q$ are $1, 1$, and $2$. Which one of the following statements is TRUE?

Both $P$ and $Q$ are diagonalizable
$P$ is diagonalizable but $Q$ is NOT diagonalizable
$P$ is NOT diagonalizable but $Q$ is diagonalizable
Both $P$ and $Q$ are NOT diagonalizable

GATE XE 2022 | Question: 6

Fri, 17 Feb 2023 06:52:34 +0000

Let $A=\left[\begin{array}{rrrr}2 & 0 & 1 & 1 \\ 1 & 2 & 5 & -5 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3\end{array}\right]$
Then the sum of the geometric multiplicities of the distinct eigenvalues of $A$ is equal to

GATE XE 2022 | Question: 9

Fri, 17 Feb 2023 06:52:32 +0000

Let $A$ and $B$ be $n \times n$ matrices with real entries.

Consider the following statements:

P: If $A$ is symmetric then $\operatorname{rank}(A)=$ Number of nonzero eigenvalues (counting multiplicity) of $A$.

Q: If $A B=\mathbf{0}$ then $\operatorname{rank}(A)+\operatorname{rank}(B) \leq n$

Then

both $\mathrm{P}$ and $\mathrm{Q}$ are TRUE
$\mathrm{P}$ is TRUE and $\mathrm{Q}$ is FALSE
$\mathrm{P}$ is FALSE and $\mathrm{Q}$ is TRUE
both $\text{P}$ and $\text{Q}$ are FALSE