Question

Assertion :If A is a square matrix of order n then $det\left(kA\right)=k^n\left|A\right|$ Reason: If matrix B is obtained from A by multiplying any row (or column) by a non zero scalar k then $det\left(B\right)=kdet\left(A\right).$

• A. Both (A) & (R) are individually true & (R) is correct explanation of (A),
• B. Both (A) & (R) are individually true but (R) is not the correct (proper) explanation of (A).
• C. (A)is true but (R} is false,
• D. (A)is false but (R} is true.

Answer 1

The correct option is A Both (A) & (R) are individually true & (R) is correct explanation of (A),

Let $\left[\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right]$

$\therefore \left|A\right|=a_{11}{a}_{22}-a_{21}{a}_{22}$ then

$kA=\left[\begin{array}{cc}k{a}_{11}& k{a}_{12}\\ k{a}_{21}& k{a}_{22}\end{array}\right]$

$\Rightarrow \left|kA\right|=\left|\begin{array}{cc}k{a}_{11}& k{a}_{12}\\ k{a}_{21}& k{a}_{22}\end{array}\right|$

$=k^2(a_{11}{a}_{22}-a_{21}{a}_{12})$ $=k^{2}detA=k^2\left|A\right|$

$\therefore$ In general if order of matrix A is, n then $\left|kA\right|=k^n\left|A\right|$

$\therefore$ Assertion (A) and Reason (R) are individually true and Reason (R) is correct explanation Of Assertion (A)