1
GATE CSE 2025 Set 2
MCQ (More than One Correct Answer)
+2
-0

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where

$$L=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}\right] \text { and } u=\left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right]$$

Which of the following statement(s) is/are TRUE?

A
The system $P X=Q$ can be solved by first solving $L Y=Q$ and then $U X=Y$.
B
If $P$ is invertible, then both $L$ and $U$ are invertible.
C
If $P$ is singular, then at least one of the diagonal elements of $U$ is zero.
D
If $P$ is symmetric, then both $L$ and $U$ are symmetric.
2
GATE CSE 2025 Set 1
MCQ (Single Correct Answer)
+2
-0

Let $A$ be a $2 \times 2$ matrix as given.

$$A=\left[\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right]$$

What are the eigenvalues of the matrix $A^{13}$ ?

A
$1,-1$
B
$2 \sqrt{2},-2 \sqrt{2}$
C
$4 \sqrt{2},-4 \sqrt{2}$
D
$64 \sqrt{2},-64 \sqrt{2}$
3
GATE CSE 2024 Set 2
MCQ (More than One Correct Answer)
+2
-0

Let A be an n × n matrix over the set of all real numbers ℝ. Let B be a matrix obtained from A by swapping two rows. Which of the following statements is/are TRUE?

A

The determinant of B is the negative of the determinant of A

B

If A is invertible, then B is also invertible

C

If A is symmetric, then B is also symmetric

D

If the trace of A is zero, then the trace of B is also zero

4
GATE CSE 2024 Set 1
MCQ (More than One Correct Answer)
+2
-0

Let A be any n x m matrix, where m > n. Which of the following statements is/are TRUE about the system of linear equations Ax = 0?

A

There exist at least m - n linearly independent solutions to this system

B

There exist m - n linearly independent vectors such that every solution is a linear combination of these vectors

C

There exists a non-zero solution in which at least m - n variables are 0

D

There exists a solution in which at least n variables are non-zero

GATE CSE Subjects
Software Engineering
Web Technologies
EXAM MAP