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(D) Given that $A$ is a $3 	imes 3$ matrix and $\operatorname{det}(A) = 4$.First,consider the matrix $2A$. Since $A$ is a $3 	imes 3$ matrix,$\opera

Vedclass · Accepted Answer

$64$

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(D) Given that $A$ is a $3 	imes 3$ matrix and $\operatorname{det}(A) = 4$.First,consider the matrix $2A$. Since $A$ is a $3 	imes 3$ matrix,$\operatorname{det}(2A) = 2^{3} 	imes \operatorname{det}(A) = 8 	imes 4 = 32$.Next,the matrix $B$ is obtained from $2A$ by the row operation $R_{2} ightarrow 2R_{2} + 5R_{3}$.Applying the property of determinants,if a row is multiplied by a scalar $k$,the determinant is multiplied by $k$. Here,$R_{2}$ is replaced by $2R_{2} + 5R_{3}$.This operation is equivalent to multiplying the second row by $2$ and then adding $5$ times the third row to it. Adding a multiple of one row to another does not change the value of the determinant.Therefore,$\operatorname{det}(B) = 2 	imes \operatorname{det}(2A) = 2 	imes 32 = 64$.

Let $A$ be a $3 \times 3$ matrix with $\operatorname{det}(A) = 4$. Let $R_{i}$ denote the $i^{\text{th}}$ row of $A$. If a matrix $B$ is obtained by performing the operation $R_{2} \rightarrow 2R_{2} + 5R_{3}$ on $2A$,then $\operatorname{det}(B)$ is equal to:

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