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

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Detailed Solution

Given the matrix A is a square matrix of order 3 and having the determinant 4.

Suppose that $A = (\begin{matrix} a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \\ c_{1} & c_{2} & c_{3} \end{matrix})$ and $2 A = (\begin{matrix} 2 a_{1} & 2 a_{2} & 2 a_{3} \\ 2 b_{1} & 2 b_{2} & 2 b_{3} \\ 2 c_{1} & 2 c_{2} & 2 c_{3} \end{matrix})$

Perform the operation $R_{2} \to 2 R_{2} + 5 R_{3}$ on the matrix $2 A$

The equivalent matrix is $2 A = (\begin{matrix} 2 a_{1} & 2 a_{2} & 2 a_{3} \\ 4 b_{1} + 10 c_{1} & 4 b_{2} + 10 c_{2} & 4 b_{3} + 10 c_{3} \\ 2 c_{1} & 2 c_{2} & 2 c_{3} \end{matrix})$

Hence,

$\begin{matrix} |2 A| = |\begin{matrix} 2 a_{1} & 2 a_{2} & 2 a_{3} \\ 4 b_{1} & 4 b_{2} & 4 b_{3} \\ 2 c_{1} & 2 c_{2} & 2 c_{3} \end{matrix}| + 0 \cdot Two rows are equal in the other matrix, so it is zero \\ = 2 (4) (2) |A| \\ = 16 (4) \\ = 64 \end{matrix}$