Algebra

Matrices & Determinants

Medium

common

If $A=\begin{bmatrix} 1 & 0 \\ -1 & 5 \end{bmatrix}$ and I is indentity matrix of order 2, then which of following statements are TRUE?

A. $A^2-6A-mI=0$ if $m=-5$ .

B. $A^2-6A+mI=0$ if $m=-5$ .

C. $A$ is a non-singular matrix.

D. $A^2=A$

Choose the correct answer from the options given below:

B and C only

A, C and D only

C and D only

A and C only

✅ Correct Option: 4

$A^2 = \begin{bmatrix} 1 & 0 \\ -6 & 25 \end{bmatrix}$ and $6A = \begin{bmatrix} 6 & 0 \\ -6 & 30 \end{bmatrix}$ , so $A^2 - 6A = -5I$ .

With $m=-5$ , statement A reads $A^2-6A+5I=0$ , which is true; statement B reads $A^2-6A-5I=0$ , which is false.

$|A| = 5 \ne 0$ , so C is true. $A^2 \ne A$ , so D is false.

2026: 21 May Shift 2

If $A = \begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}$ such that $A^2 + \beta A + \gamma I = 0$ then $\beta + \gamma$ is equal to:

(where $I$ is an identity matrix of order 2)

2025: 14 May Shift 2

If A is a square matrix of order 3 such that $|A| = 3$ , then $|adj(adj A)|$ is equal to:

2026: 29 May Shift 2

If $A+B = \begin{bmatrix}7 & 0\\2 & 5\end{bmatrix}$ and $A-B = \begin{bmatrix}3 & 0\\0 & 3\end{bmatrix}$ , then

2026: 21 May Shift 2

If A is a matrix of order $3 \times 3$ such that $|A| = 4$ and $|k \cdot adj\, A| = 32|A|, k \in \mathbb{R}$ ; then the value of k is:

2026: 22 May Shift 1

If $A=\begin{bmatrix}\lambda & 0 & 0\\0 & \lambda & 0\\0 & 0 & \lambda\end{bmatrix}$ , $\lambda\ne0$ , then $|adjA|$ is equal to

2026: 22 May Shift 2

Match List-I with List-II

List-I	List-II
(A) If $A=\begin{bmatrix}2 & -5\\3 & 4\end{bmatrix}$ , then adj A =	(I) $\begin{bmatrix}4 & -5\\-2 & -3\end{bmatrix}$
(B) If $A=\begin{bmatrix}2 & 3\\5 & -2\end{bmatrix}$ , then $A^{-1}=$	(II) $\begin{bmatrix}\frac{1}{17} & \frac{-5}{17}\\\frac{3}{17} & \frac{2}{17}\end{bmatrix}$
(C) If $A=\begin{bmatrix}-3 & 5\\2 & 4\end{bmatrix}$ , then adj A =	(III) $\begin{bmatrix}4 & 5\\-3 & 2\end{bmatrix}$
(D) If $A=\begin{bmatrix}2 & 5\\-3 & 1\end{bmatrix}$ , then $A^{-1}=$	(IV) $\begin{bmatrix}\frac{2}{19} & \frac{3}{19}\\\frac{5}{19} & \frac{-2}{19}\end{bmatrix}$

Choose the correct answer from the options given below:

Solution