Assertion (A) & Reason (B) MCQ

Question 11 Mark

Assertion (A) : The inverse of a matrix $A=\left(\begin{array}{lll}43 & 1 & 6 \\ 35 & 7 & 4 \\ 17 & 3 & 2\end{array}\right)$ does not exist.
Reason (R) : The inverse of singular matrix is not possible.

Answer

(a) : $|A|=\left|\begin{array}{lll}43 & 1 & 6 \\ 35 & 7 & 4 \\ 17 & 3 & 2\end{array}\right|$
Applying $C_1 \rightarrow C_1-7 C_3$, we get
$
|A|=\left|\begin{array}{lll}
1 & 1 & 6 \\
7 & 7 & 4 \\
3 & 3 & 2
\end{array}\right|=0 \quad\left(\because C_1 \sim C_2\right)
$
$\therefore \quad A$ is singular matrix
$\because \quad A$ is singular matrix $\therefore A^{-1}$ does not exist.
$\therefore \quad$ Assertion and reason both are true and reason is the correct explanation of assertion.

View full question & answer→

Question 21 Mark

Assertion (A) : The inverse of the matrix $A=\left(\begin{array}{ccc}4 & 2 & 3 \\ 8 & 5 & 2 \\ 12 & -4 & 5\end{array}\right)$ certainly exists.
Reason (R) : The matrix $A$ is non-singular and every non-singular matrix possesses its inverse.

Answer

(a) : We have $A=\left(\begin{array}{ccc}4 & 2 & 3 \\ 8 & 5 & 2 \\ 12 & -4 & 5\end{array}\right)$
$
\begin{aligned}
\therefore \quad|A| & =4(25+8)-2(40-24)+3(-32-60) \\
& =-176 \neq 0
\end{aligned}
$
$\Rightarrow A$ is non-singular $\Rightarrow A^{-1}$ exists.
$\therefore$ Assertion and reason both are correct and reason is the correct explanation of assertion.

View full question & answer→

Question 31 Mark

Assertion (A) : The inverse of $A=\left(\begin{array}{ll}3 & 4 \\ 3 & 5\end{array}\right)$ does not exist.
Reason (R) : The matrix $A$ is non-singular.

Answer

(a) : $\because|A|=\left|\begin{array}{ll}3 & 4 \\ 3 & 5\end{array}\right|=15-12=3 \neq 0$
$\therefore \quad A$ is non-singular.
$\therefore \quad A^{-1}$ exists.

View full question & answer→

Question 41 Mark

Assertion (A) : The inverse of the matrix $\left[\begin{array}{ccc}1 & 3 & 5 \\ 2 & 6 & 10 \\ 9 & 8 & 7\end{array}\right]$ does not exist.
Reason (R) : The matrix $\left[\begin{array}{ccc}1 & 3 & 5 \\ 2 & 6 & 10 \\ 9 & 8 & 7\end{array}\right]$ is singular.

Answer

(a) : $\because\left[\begin{array}{ccc}1 & 3 & 5 \\ 2 & 6 & 10 \\ 9 & 8 & 7\end{array}\right]$ is singular therefore inverse does not exist.
$
\left[\because\left|\begin{array}{ccc}
1 & 3 & 5 \\
2 & 6 & 10 \\
9 & 8 & 7
\end{array}\right|=0 \text {, since } R_2=2 R_1\right]
$
$\therefore \quad$ Assertion and reason are both true but reason is the correct explanation of assertion.

View full question & answer→

Question 51 Mark

Assertion (A) : If $A=\left(\begin{array}{ccc}l & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & n\end{array}\right)$, then
$
A^{-1}=\left(\begin{array}{ccc}
1 / l & 0 & 0 \\
0 & 1 / m & 0 \\
0 & 0 & 1 / n
\end{array}\right)
$
Reason $( R )$ : The inverse of a diagonal matrix is a diagonal matrix.

Answer

(b) : $\because A=\left(\begin{array}{ccc}l & 0 & 0 \\ 0 & m & 0 \\ 0 & 0 & n\end{array}\right)$
$\therefore \quad|A|=l m n$ and $\operatorname{adj}(A)=\left(\begin{array}{ccc}m n & 0 & 0 \\ 0 & l n & 0 \\ 0 & 0 & l m\end{array}\right)$
$
\therefore \quad A^{-1}=\frac{\operatorname{adj} A}{|A|}=\left(\begin{array}{ccc}
1 / l & 0 & 0 \\
0 & 1 / m & 0 \\
0 & 0 & 1 / n
\end{array}\right)=\operatorname{diag}\left(\frac{1}{l}, \frac{1}{m}, \frac{1}{n}\right)
$
$\therefore \quad$ Assertion and reason are both correct but reason is not the correct explanation of assertion.

View full question & answer→

Question 61 Mark

Assertion (A) : If $A$ is skew-symmetric of order 3, then its determinant should be zero.
Reason (R) : If $A$ is square matrix, then $\operatorname{det} A=\operatorname{det} A^{\prime}=\operatorname{det}\left(-A^{\prime}\right)$.

Answer

(b) : Reason is false since
$\operatorname{det} A^{\prime}=\operatorname{det}\left(-A^{\prime}\right)$ is not true.
Indeed, $\operatorname{det}\left(-A^{\prime}\right)=(-1)^3 \operatorname{det} A^{\prime}$
Now as $A=-A^{\prime} \quad(\because A$ is skew-symmetric $)$
$\therefore \quad \operatorname{det} A=\operatorname{det}\left(-A^{\prime}\right)=-\operatorname{det}\left(A^{\prime}\right)=-\operatorname{det} A$
$\Rightarrow \operatorname{det} A=0$.
Thus assertion is correct.
$\therefore \quad$ Assertion and reason are both correct but reason is not the correct explanation of assertion.

View full question & answer→

Question 71 Mark

Let $A$ be a $2 \times 2$ matrix.
Assertion $( A ): \operatorname{adj}(\operatorname{adj} A)=A$.
Reason (R): $|\operatorname{adj} A|=|A|$.

Answer

(b) : Let $A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]$, adj $A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right]$
$\Rightarrow \operatorname{adj}(\operatorname{adj} A)=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=A$
$\operatorname{adj} A=\left[\begin{array}{cc}d & -b \\ -c & a\end{array}\right] \Rightarrow|\operatorname{adj} A|=a d-b c=|A|$.

View full question & answer→

Question 81 Mark

Let $A=\left[\begin{array}{ccc}1 & 0 & a \\ 2 & 3 & b \\ -3 & 1 & c\end{array}\right], B=\left[\begin{array}{ccc}1 & 0 & x \\ 2 & 3 & y \\ -3 & 1 & z\end{array}\right]$
and $C=\left[\begin{array}{ccc}1 & 0 & a+x \\ 2 & 3 & b+y \\ -3 & 1 & c+z\end{array}\right]$.
Assertion (A) : $\operatorname{det} A+\operatorname{det} B=\operatorname{det} C$.
Reason (R) : $A+B=C$.

Answer

(c) : Clearly $A+B \neq C$. Hence, reason is wrong.
However, by a property of determinants, $\operatorname{det} C=\operatorname{det} A+\operatorname{det} B$.
$\therefore \quad$ Assertion is correct statement but reason is wrong statement.

View full question & answer→

Question 91 Mark

Assertion (A) : If $A=\left(\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right)$, then $\operatorname{adj}(\operatorname{adj} A)=A$.
Reason (R) : $|\operatorname{adj}(\operatorname{adj} A)|=|A|^{(n-1)^2}, A$ be $n$ rowed non singular matrix.

Answer

(b): $\operatorname{adj}(\operatorname{adj} A)=|A|^{n-2} A$
Here, $n=3$
$\therefore \quad \operatorname{adj}(\operatorname{adj} A)=|A| A$ ....(i)
Now, $|A|=\left|\begin{array}{lll}3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1\end{array}\right|=3(-3+4)+3(2)+4(-2)=1$
From Eq. (i), adj $(\operatorname{adj} A)=A$

View full question & answer→

Question 101 Mark

Assertion (A) : The determinant of $A=\left(\begin{array}{lll}1 & 2 & 3 \\ 5 & 6 & 7 \\ 2 & 4 & 6\end{array}\right)$ is zero.
Reason (R) : The determinant of a matrix vanishes if any two rows or columns are proportional.

Answer

(a) : $|A|=\left|\begin{array}{lll}1 & 2 & 3 \\ 5 & 6 & 7 \\ 2 & 4 & 6\end{array}\right|=\left|\begin{array}{ccc}4 & 2 & 3 \\ 12 & 6 & 7 \\ 8 & 4 & 6\end{array}\right|$
$
\text { (using } C_1 \rightarrow C_1+C_3 \text { ) }
$
$\therefore \quad|A|=0\left(\because C_1 \sim C_2\right)$
$\therefore \quad$ Assertion and reason are both correct and reason is the correct explanation of assertion.

View full question & answer→

MATHS Assertion (A) & Reason (B) MCQ

Take a timed test