Question

If A is a $3\times 3$ matrix and |3A| = k |A|, then k =

• A. 9
• B. 6
• C. 1
• D. 27

Answer 1

The correct option is D 27

Since$A$ is a $3\times 3$ matrix, $A$ has $3$ rows.

$3A$ is the matrix obtained by multiplying each entry of $A$ by $3.$

Thus if $A$ has row vectors $a_1,a_2$ and $a_3$.

$3A$ has row vectors $3a_1,3a_2$ and $3a_3$.

Since multiplying a single row of a matrix $A$ by $a$ scalar $r$ has the effect of multiplying the determinant of $A$ by $r$, we obtain:

det $\left(3A\right)=3\times 3\times 3$ det $\left(A\right)$$=27$ det $\left(A\right)$

$\therefore k=27$.