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Matrices Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Matrices Set-6

Practice Important MCQs for Matrices — Mathematics, Class 12.

Matrices form the backbone of many core ideas in Class 12 Mathematics, such as rank, determinants, eigenvalues, and transformations. They also appear frequently in board exams and competitive tests (CBSE, JEE, NEET) because they provide a compact way to solve systems of linear equations, analyze invertibility, and characterize linear transformations.

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Q1. Let

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

For which integer values of $k$ does $A^{-1}$ have all integer entries?

(A)

$\{5\}$

(B)

$\{7\}$

(C)

$\{5,7\}$

(D)

No integer value of $k$

Q2. Let

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

For which value of $k$ is $\operatorname{rank}(A)=2$ ?

(A)

$k=5$

(B)

$k=3$

(C)

$k=1$

(D)

$k=0$

Q3. Let $A$ be a non-zero $3\times 3$ real matrix such that $A^2=0$ (the zero matrix). What is $\operatorname{rank}(A)$ ?

(A)

$0$

(B)

$3$

(C)

$2$

(D)

$1$

Q4. Let $A$ be a $3\times3$ real matrix satisfying $A^2=A$ and $\operatorname{rank}(A)=2$ . Which of the following must be true?

(A)

$\det(A)=1$

(B)

The minimal polynomial of $A$ is $x(x-1)$

(C)

$A$ is invertible

(D)

All eigenvalues of $A$ are $1$

Q5. For which value of $k$ does the following system have infinitely many solutions?

\begin{aligned} x+y+z &= 3 <br> 2x+ky+4z &= 6 <br> 3x+(k+1)y+7z &= 9 \end{aligned}

(A)

$k=2$

(B)

$k=1$

(C)

$k=3$

(D)

The system has a unique solution for every real $k$

Q6. Let $A$ be an invertible $3\times3$ matrix whose eigenvalues are $1,\;2,\;4$ . Find $\det\bigl(2A^{-1}+I\bigr)$ .

(A)

$1$

(B)

$9$

(C)

$8$

(D)

$24$

Q7. If $A$ is a $3\times3$ real matrix with $\operatorname{rank}(A)=1$ and $\operatorname{tr}(A)=5$ , then $\det(I+A)=$

(A)

$5$

(B)

$0$

(C)

$1$

(D)

$6$

Q8. Let $A$ be a $3\times3$ real matrix satisfying $A^2=A^{T}$ . The possible values of $\det(A)$ are

(A)

$0\ \text{or}\ 1$

(B)

$-1$

(C)

$1$

(D)

$0$

Q9. Let $A$ be an invertible $3\times3$ matrix satisfying $A^3-3A+2I=0$ . Then $A^{-1}=$

(A)

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

(B)

$\dfrac{3I+A^2}{2}$

(C)

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

(D)

$\dfrac{A^2-I}{2}$

Q10. Let $A$ be a non-zero $3\times3$ real matrix such that $A^2=0$ (the zero matrix). Then $\operatorname{rank}(A)=$

(A)

$0$

(B)

$1$

(C)

$2$

(D)

$3$

Q11. Let $A$ be a $2\times2$ real matrix satisfying $A^2=9I_2$ and $\operatorname{tr}(A)=0$ . Then $\det(A)=$

(A)

$9$

(B)

$0$

(C)

$-9$

(D)

$81$

Q12. Let $M=\begin{pmatrix}a & 1 & 1\$$4pt]1 & a & 1\$$4pt]1 & 1 & a\end{pmatrix}$ . Then $\det(M)=$ and $M$ is invertible for which $a$ ?

(A)

$\det(M)=(a+2)(a-1)^2$ , invertible iff $a\neq1,\,-2$

(B)

$\det(M)=(a-1)^3$ , invertible iff $a\neq1$

(C)

$\det(M)=(a+1)^2(a-2)$ , invertible iff $a\neq-1,\,2$

(D)

$\det(M)=(a+2)(a-1)$ , invertible iff $a\neq1,\,-2$

Q13. Let $A$ be a real symmetric $3\times3$ matrix with eigenvalues $1,2,3$ . Then $\operatorname{tr}\big(\operatorname{adj}(A)\big)=$

(A)

$6$

(B)

$\tfrac{7}{4}$

(C)

$14$

(D)

$11$

Q14. Let $A$ be a $3\times3$ real matrix such that $\operatorname{adj}(A)=I_3$ . Which of the following is necessarily true?

(A)

$\det(A)=1$

(B)

$A=\pm I_3$

(C)

$A$ is invertible with $A^{-1}=2\det(A)\,I_3$

(D)

$\operatorname{tr}(A)=3$

Q15. Let $A$ be a $4\times4$ real matrix with $A^2=0$ and $A\neq0$ . Which statement must be true?

(A)

$\operatorname{rank}(A)=3$

(B)

$\det(A)\neq0$

(C)

$\operatorname{rank}(A)\leq2$

(D)

$\dim\ker(A)=1$

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