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Matrices Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Matrices Set-4

Practice Important MCQs for Matrices — Mathematics, Class 12.

Matrices form the backbone of many ideas in Class 12 Mathematics and are frequently used in board as well as competitive examinations—especially for concepts like determinants, inverses, rank, eigenvalues/diagonalization links, and systems of linear equations. Mastery of matrix properties and related theorems helps you solve a wide range of problems quickly and accurately.

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Q1. Let $A=\begin{pmatrix}k+1 & 2 \$$4pt] 3 & k-1\end{pmatrix}$ . For which real values of $k$ is $A$ invertible?

(A)

$k\neq 0$

(B)

$k=\pm\sqrt{7}$

(C)

$k\neq\pm\sqrt{7}$

(D)

$k\in\{-1,1\}$

Q2. Let $A$ and $B$ be $3\times3$ real matrices with $\operatorname{rank}(A)=2$ and $AB=0$ (zero matrix). What is the maximum possible value of $\operatorname{rank}(B)$ ?

(A)

$1$

(B)

$2$

(C)

$0$

(D)

$3$

Q3. Consider the system

\begin{aligned} x+y+z &= 3 <br> 2x+ky+4z &= 8 <br> x+2y+(k+1)z &= 4 \end{aligned}

For which real value(s) of $k$ does the system have infinitely many solutions?

(A)

$k=1\pm\sqrt{3}$

(B)

\text{No real value of } $k$

(C)

$k=1$

(D)

$k=-1\pm\sqrt{3}$

Q4. Assertion (A): For an $n\times n$ real matrix $A$ , if $\operatorname{rank}(A)=n-1$ then $\operatorname{adj}(A)$ is the zero matrix.
Reason (R): If $\operatorname{rank}(A)\le n-2$ then $\operatorname{adj}(A)$ is the zero matrix.

(A)

Both A and R are true and R is the correct explanation of A.

(B)

Both A and R are true but R is not the correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q5. Let $A$ be a $3\times3$ real matrix satisfying $A^2=0$ (zero matrix). Which of the following is correct about possible ranks of $A$ and of $I_3+A$ ?

(A)

$\operatorname{rank}(A)\in\{0,1\}\ \text{and}\ \operatorname{rank}(I_3+A)=3$

(B)

$\operatorname{rank}(A)\in\{0,1,2\}\ \text{and}\ \operatorname{rank}(I_3+A)=3$

(C)

$\operatorname{rank}(A)\in\{0,1\}\ \text{and}\ \operatorname{rank}(I_3+A)\in\{2,3\}$

(D)

$\operatorname{rank}(A)=0\ \text{and}\ \operatorname{rank}(I_3+A)=3$

Q6. For which value of $p$ is the matrix

A=\begin{pmatrix}1 & 2 & 3 \$$4pt] 2 & 5 & 3 \$$4pt] 3 & 3 & p\end{pmatrix}

singular?

(A)

$p=0$

(B)

$p=9$

(C)

$p=18$

(D)

$p=-18$

Q7. For which value(s) of $k$ does the system

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

have infinitely many solutions?

(A)

$k=1\text{ or }k=3$

(B)

$k=2$

(C)

$k=-1$

(D)

No value of $k$ gives infinitely many solutions

Q8. Let $A$ be a non-zero $3\times3$ real matrix satisfying $A^2=0$ . Then $\operatorname{rank}(A)$ equals

(A)

$0$

(B)

$2$

(C)

$3$

(D)

$1$

Q9. Let $A$ be a real $3\times3$ matrix such that $\operatorname{rank}(A-I)=\operatorname{rank}(A+I)=1$ . Which of the following is true?

(A)

$\det(A)=1$

(B)

No such matrix exists

(C)

$\det(A)=-1$

(D)

$\det(A)=0$

Q10. If $A$ and $B$ are $3\times3$ real matrices with $\operatorname{rank}(A)=\operatorname{rank}(B)=2$ , which of the following can be $\operatorname{rank}(AB)$ ?

(A)

$0$ only

(B)

$1$ only

(C)

$2$ only

(D)

$1\text{ or }2$

Q11. Let $A=\displaystyle\begin{pmatrix}2 & 3\$$4pt]1 & k\end{pmatrix}$ . For what value of $k$ does $A^{-1}$ have equal diagonal entries?

(A)

$k=\dfrac{3}{2}$

(B)

$k=2$

(C)

$k=-2$

(D)

$k=1$

Q12. Let $A=\displaystyle\begin{pmatrix}a & 1 & 1\$$4pt]1 & a & 1\$$4pt]1 & 1 & a\end{pmatrix}$ where $a\in\mathbb{R}$ . The rank of $A$ is:

(A)

rank $3$ for all real $a$ and rank $0$ when $a=1$

(B)

rank $2$ if $a=1$ , rank $1$ if $a=-2$ , and rank $3$ otherwise

(C)

rank $1$ if $a=1$ , rank $3$ if $a=-2$ , and rank $2$ otherwise

(D)

rank $1$ if $a=1$ , rank $2$ if $a=-2$ , and rank $3$ for all other real $a$

Q13. Let $A=\displaystyle\begin{pmatrix}1 & k\$$4pt]2 & 3\end{pmatrix}$ with $k\in\mathbb{Z}$ . For which integer values of $k$ does $A^{-1}$ have all integer entries?

(A)

$k=1$ or $k=2$

(B)

$k=0$ or $k=3$

(C)

$k=1$ only

(D)

$k=2$ only

Q14. Assertion (A): For a $3\times3$ real matrix $A$ , if $\operatorname{rank}(A)=2$ then $\operatorname{adj}(A)$ is a non-zero matrix of rank $1$ .
Reason (R): For any $n\times n$ matrix $A$ , $\operatorname{rank}(\operatorname{adj}(A))=1$ if $\operatorname{rank}(A)=n-1$ , and $\operatorname{adj}(A)=0$ if $\operatorname{rank}(A)\le n-2$ .

(A)

Both A and R are true, but R is not a correct explanation of A

(B)

A is true and R is false

(C)

Both A and R are true and R is a correct explanation of A

(D)

A is false but R is true

Q15. Let $A$ be a $3\times3$ real matrix satisfying $A^{2}-A-I=0$ . Which of the following is necessarily true?

(A)

$A$ is singular

(B)

$A$ is invertible and $A^{-1}=A-I$

(C)

$A$ is invertible and $A^{-1}=I-A$

(D)

$\det(A)=1$

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