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Matrices Set-5

Practice Important MCQs for Matrices — Mathematics, Class 12.

Matrices are central to CBSE Class 12 and competitive exams because they provide a systematic language to represent linear transformations and solve systems of equations efficiently. Concepts like determinants, adjoint (adjugate), inverse, rank, eigenvalues, and properties of special matrices frequently appear in board questions and form the backbone of tougher JEE/NEET-style problems.

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Q1. If $A=\begin{pmatrix}1 & 2 \$$4pt] 3 & k\end{pmatrix}$ and $A^{-1}=\dfrac{1}{5}\begin{pmatrix}k & -2 \$$4pt] -3 & 1\end{pmatrix}$ , then $k=$ ?

(A)

$9$

(B)

$10$

(C)

$11$

(D)

$-1$

Q2. For which value of $k$ does the system

\begin{aligned} x+2y+3z&=1 \$$4pt] 2x+3y+z&=2 \$$4pt] 3x+5y+kz&=3 \end{aligned}

have infinitely many solutions?

(A)

$4$

(B)

$3$

(C)

$5$

(D)

$2$

Q3. For which value of $k$ is the matrix

A=\begin{pmatrix}1 & 1 & 1 \$$4pt] 1 & 2 & 3 \$$4pt] 1 & 3 & k\end{pmatrix}

singular?

(A)

$1$

(B)

$5$

(C)

$-1$

(D)

$\dfrac{2}{3}$

Q4. Let $A$ be a $3\times3$ real matrix with $\det(A)=0$ and $\operatorname{adj}(A)\neq 0$ . If the system $Ax=b$ is consistent, then:

(A)

it has a unique solution

(B)

it has no solution

(C)

it has exactly two solutions

(D)

it has infinitely many solutions

Q5. Let $A$ and $B$ be $2\times2$ real matrices. Which one of the following statements is true?

(A)

There do not exist $2\times2$ real matrices $A,B$ such that $AB-BA=I_2$ .

(B)

If $AB-BA=0$ then $A$ is invertible.

(C)

$\det(AB-BA)=0$ for all $A,B$ .

(D)

$AB-BA$ is always nilpotent for all $A,B$ .

Q6. Consider the matrix $A=\begin{pmatrix}k & 1 & 1 \$$2pt] 1 & k & 1 \$$2pt] 1 & 1 & k \end{pmatrix}$ . For which values of $k$ is $A$ singular?

(A)

$k=1$ only

(B)

$k=1\ \text{or}\ k=-2$

(C)

$k=-2$ only

(D)

$k=0\ \text{or}\ k=1$

Q7. Let $A$ be an invertible $3\times3$ matrix whose eigenvalues are $1,2,3$ . The value of $\operatorname{tr}\big(\operatorname{adj}(A^2)\big)$ is

(A)

$18$

(B)

$14$

(C)

$71$

(D)

$49$

Q8. Let $A$ be an invertible $n\times n$ matrix and $B$ an $n\times n$ matrix such that $ABA^{-1}=3B$ . Which of the following must necessarily be true?

(A)

$\operatorname{tr}(B)=0$

(B)

$B=0$

(C)

$\det(B)=0$

(D)

$B$ is diagonalizable

Q9. Assertion (A): If $A$ is a real skew-symmetric matrix ( $A^{T}=-A$ ) of order $n$ , then $\operatorname{tr}(A)=0$ .
Reason (R): All eigenvalues of a real skew-symmetric matrix are $0$ .

(A)

Both (A) and (R) are true and (R) explains (A)

(B)

Both (A) and (R) are true but (R) does not explain (A)

(C)

(A) is true but (R) is false

(D)

(A) is false but (R) is true

Q10. Let $A$ be an $n\times n$ matrix with $n\ge2$ and $\operatorname{rank}(A)=n-1$ . Which of the following statements about $\operatorname{adj}(A)$ is always true?

(A)

$\operatorname{adj}(A)=0$

(B)

$\operatorname{rank}\big(\operatorname{adj}(A)\big)=1$

(C)

$\operatorname{adj}(A)$ is invertible

(D)

$\operatorname{rank}\big(\operatorname{adj}(A)\big)=n-1$

Q11. If $A=\begin{pmatrix}2 & 1 \$$4pt] 3 & 2\end{pmatrix}$ , then $A^{-1}=$

(A)

$\begin{pmatrix}2 & -1 \$$4pt] -3 & 1\end{pmatrix}$

(B)

$\begin{pmatrix}2 & -1 \$$4pt] -3 & 2\end{pmatrix}$

(C)

$\begin{pmatrix}2 & 1 \$$4pt] -3 & 2\end{pmatrix}$

(D)

$\begin{pmatrix}1 & -1 \$$4pt] -3 & 2\end{pmatrix}$

Q12. For which value of $k$ does the matrix $A=\begin{pmatrix}1 & 2 & 3 \$$4pt] 2 & 5 & 8 \$$4pt] 3 & 8 & k\end{pmatrix}$ have rank $2$ ?

(A)

$k=-13$

(B)

$k=0$

(C)

$k=8$

(D)

$k=13$

Q13. For which value of $k$ does the linear system

\begin{aligned} x+y+z &= 1 \$$4pt] 2x+3y+2z &= k \$$4pt] 3x+4y+3z &= 2 \end{aligned}

have infinitely many solutions?

(A)

$k=1$

(B)

$k=-1$

(C)

$k=2$

(D)

No real $k$ gives infinitely many solutions

Q14. Let $A$ be an $n\times n$ real matrix. Does there exist an $n\times n$ real matrix $X$ such that $AX - XA = I_n$ ?

(A)

Yes, for every $n\ge 1$

(B)

Yes, but only when $n$ is even

(C)

No, such an $X$ does not exist for any $n\ge 1$

(D)

Yes, but only when $n$ is prime

Q15. Let $A$ be a $3\times 3$ real matrix satisfying $\operatorname{adj}(A)=A^{T}$ . Then $\det(A)=$

(A)

$1$

(B)

$0\ \text{or}\ 1$

(C)

$-1$

(D)

$\pm 1$

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