Question

If A is a square matrix of order $3\times 3$, then |KA| is equal to

• A. $K^2|A|$
• B. $K^3|A|$
• C. $K|A|$
• D. $3K|A|$

Answer 1

Answer: (B)

$K^3|A|$

A is a square matrix of order 3x3
$\left|KA\right|=K^N\left|A\right|$
$=K^3\left|A\right|$
If a matrix is multiplied by scalar then all elements of matrix get multiplied
But if a determinant is multiplied by scalar, only one row or column gets multiplied
$\therefore \left|KA\right|=\left|\begin{array}{ccc}K{a}_{11}& K{a}_{12}& K{a}_{13}\\ K{a}_{21}& K{a}_{22}& K{a}_{23}\\ K{a}_{31}& K{a}_{32}& K{a}_{33}\end{array}\right|$
$=K\cdot K\cdot K\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|$
$=K^3\left|A\right|$