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Matrices Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Matrices Set-3

Practice Important MCQs for Matrices — Mathematics, Class 12.

Matrices are a core topic in Class 12 Mathematics because they provide a compact and powerful way to study systems of linear equations, transformations, and linear independence. They are heavily used in board exams as well as competitive exams (JEE/NEET) for quickly solving problems involving ranks, determinants, eigen-properties, and inverses—so mastering them directly improves both speed and accuracy.

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Q1. (Let $A$ be a $2\times2$ invertible matrix with $\det(A)=-4$ . Find $\det\!\big(3A^{-1}\big)$ .)

(A)

$-\dfrac{3}{4}$

(B)

$\dfrac{9}{4}$

(C)

$-\dfrac{9}{4}$

(D)

$-\dfrac{9}{2}$

Q2. (For which real value(s) of $k$ does the system

\begin{aligned} x+2y-z &= 1 <br> 2x+5y-3z &= 2 <br> x+ky-2z &= 0 \end{aligned}

have infinitely many solutions?)

(A)

No real value of $k$

(B)

$k=3$

(C)

$k=1$

(D)

All real numbers

Q3. (If $A$ is a real $3\times3$ matrix satisfying $A^{T}A=2I_3$ , then $\det(A)$ equals:)

(A)

$2\sqrt{2}$

(B)

$-2\sqrt{2}$

(C)

$0$

(D)

$\pm\,2\sqrt{2}$

Q4. (For which positive integer $n$ do there exist $n\times n$ real matrices $A,B$ such that $AB-BA=I_n$ ?)

(A)

Only for even $n$

(B)

No positive integer $n$ (no such matrices exist)

(C)

Only for odd $n$

(D)

Only for $n=1$

Q5. (Let $A$ be a $3\times3$ real matrix satisfying $A^2=A^{T}$ . Then $\det(A)$ must be:)

(A)

$0$

(B)

$1$

(C)

$0\ \text{or}\ 1$

(D)

Any real number

Q6. Let $A$ be a $3\times3$ matrix satisfying $A^2-5A+6I=0$ and $A$ is invertible. Then $A^{-1}=$

(A)

$A^{-1}=\dfrac{A-5I}{6}$

(B)

$A^{-1}=\dfrac{3I-A}{2}$

(C)

$A^{-1}=\dfrac{5I-A}{6}$

(D)

$A^{-1}=\dfrac{A-I}{6}$

Q7. Let $A=\begin{pmatrix}k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k\end{pmatrix}$ . For which values of $k$ is $A$ invertible and what is $A^{-1}$ in closed form (write your answer using $I_3$ and $J_3$ , the $3\times3$ identity and all-ones matrices respectively)?

(A)

$A$ is invertible for $k\neq 1,-2$ and $A^{-1}=\dfrac{1}{k-1}I_3-\dfrac{1}{(k-1)(k+2)}J_3$

(B)

$A$ is invertible for $k\neq 1,-2$ and $A^{-1}=\dfrac{1}{k-1}I_3+\dfrac{1}{(k-1)(k+2)}J_3$

(C)

$A$ is invertible for $k\neq -1,2$ and $A^{-1}=\dfrac{1}{k+2}I_3-\dfrac{1}{(k-1)(k+2)}J_3$

(D)

$A$ is invertible for $k\neq 1,-2$ and $A^{-1}=\dfrac{1}{k+2}I_3-\dfrac{1}{(k-1)(k+2)}J_3$

Q8. Consider the system of linear equations

\begin{aligned} x+2y+3z&=4\\ 2x+4y+6z&=k\\ x+y+z&=1 \end{aligned}

For which value(s) of $k$ does the system have no solution, infinitely many solutions, or a unique solution?

(A)

Unique solution for all $k\neq 8$ ; no solution for $k=8$

(B)

Unique solution for $k=8$ ; no solution otherwise

(C)

Infinitely many solutions for every real $k$

(D)

No solution when $k\neq 8$ , infinitely many solutions when $k=8$ , and never a unique solution

Q9. Let $A$ be a real $3\times3$ matrix with $A^3=I$ , $A\neq I$ , and $\det(A)=1$ . Then $\operatorname{tr}(A)=$

(A)

$3$

(B)

$0$

(C)

$-1$

(D)

$1$

Q10. Let $M(x)$ be the $4\times4$ matrix whose diagonal entries are $x$ and every off-diagonal entry is $1$ (i.e. all diagonal entries $x$ , all off-diagonal entries $1$ ). Which statement is correct about $\det M(x)$ and singular values of $M(x)$ ?

(A)

$\det M(x)=(x+3)^3(x-1)$ , so $M(x)$ is singular at $x=-3$ (multiplicity $3$ ) and $x=1$ (multiplicity $1$ )

(B)

$\det M(x)=(x+3)^2(x-1)^2$ , so $M(x)$ is singular at $x=-3$ and $x=1$ (each with multiplicity $2$ )

(C)

$\det M(x)=(x+3)(x-1)^3$ , eigenvalues are $x+3$ (simple) and $x-1$ (multiplicity $3$ ), so singular at $x=-3$ (mult.\ $1$ ) and $x=1$ (mult.\ $3$ )

(D)

$\det M(x)=(x+1)(x-1)^3$ , so $M(x)$ is singular at $x=-1$ (mult.\ $1$ ) and $x=1$ (mult.\ $3$ )

Q11. Let $A$ be an invertible $3\times3$ matrix with $\det(A)=5$ . Find $\det\bigl(3A^{-1}\bigr)$ .

(A)

$\dfrac{5}{27}$

(B)

$\dfrac{27}{5}$

(C)

$-\dfrac{27}{5}$

(D)

$135$

Q12. Consider the system of linear equations

\begin{aligned} x + 2y + 3z &= 1 <br> 2x + 5y + 8z &= 2 <br> 3x + 8y + kz &= 3 \end{aligned}

For which value of $k$ does the system have infinitely many solutions?

(A)

$k=0$

(B)

$k=-13$

(C)

All real $k$

(D)

$k=13$

Q13. Let $A=\begin{pmatrix}1 & 2 & 3 \$$4pt] k & 4 & 6 \$$4pt] 2k & 8 & 12\end{pmatrix}$ . For which value(s) of $k$ does $\operatorname{rank}(A)=1$ ?

(A)

$k=2$

(B)

$k=0$

(C)

$k=-2$

(D)

No real value of $k$

Q14. If $A$ is a non-zero $3\times3$ real matrix satisfying $A^2=O$ , which of the following describes the possible values of $\operatorname{rank}(A)$ ?

(A)

$\{0\}$

(B)

$\{2\}$

(C)

$\{1\}$

(D)

$\{1,2\}$

Q15. Let $A$ be a $3\times3$ real matrix such that $\operatorname{adj}(A)=A$ . Then the possible values of $\det(A)$ are

(A)

$\{1\}$

(B)

$\{0,1\}$

(C)

$\{0\}$

(D)

$\{-1,1\}$

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