1
GATE CE 2025 Set 1
MCQ (Single Correct Answer)
+1
-0.33

Suppose $\lambda$ is an eigenvalue of matrix A and $x$ is the corresponding eigenvector. Let $x$ also be an eigenvector of the matrix $\mathrm{B}=\mathrm{A}-2 \mathrm{I}$, where I is the identity matrix. Then, the eigenvalue of B corresponding to the eigenvector $x$ is equal to

A
$\lambda$
B
$\lambda+2$
C
$2 \lambda$
D
$\lambda-2$
2
GATE CE 2025 Set 1
MCQ (Single Correct Answer)
+1
-0.33

Let $A=\left[\begin{array}{cc}1 & 1 \\ 1 & 3 \\ -2 & -3\end{array}\right]$ and $b=\left[\begin{array}{l}b_1 \\ b_2 \\ b_3\end{array}\right]$. For $\mathrm{Ax}=\mathrm{b}$ to be solvable, which one of the following options is the correct condition on $b_1, b_2$ and $b_3$ :

A
$b_1+b_2+b_3=1$
B
$3 b_1+b_2+2 b_3=0$
C
$b_1+3 b_2+b_3=2$
D
$b_1+b_2+b_3=2$
3
GATE CE 2024 Set 2
MCQ (Single Correct Answer)
+1
-0.33

The statements P and Q are related to matrices A and B, which are conformable for both addition and multiplication.

P: $(A + B)^T = A^T + B^T$

Q: $(AB)^T = B^T A^T$

Which one of the following options is CORRECT?

A

P is TRUE and Q is FALSE

B

Both P and Q are TRUE

C

P is FALSE and Q is TRUE

D

Both P and Q are FALSE

4
GATE CE 2023 Set 2
MCQ (More than One Correct Answer)
+1
-0

For the matrix

$\rm [A]=\begin{bmatrix}1&-1&0\\\ -1&2&-1\\\ 0&-1&1\end{bmatrix}$

which of the following statements is/are TRUE?

A
[𝐴]{𝑥} = {𝑏} has a unique solution
B
[𝐴]{𝑥} = {𝑏} does not have a unique solution 
C
[𝐴] has three linearly independent eigenvectors
D
[𝐴] is a positive definite matrix
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