Question

If $A=\left[a_{ij}\right]$ is a $4\times 4$ matrix and $c_{ij}$ is the co-factor of the element $a_{ij}$ in $\left|A\right|$, then the expression $a_{11}{c}_{11}+a_{12}{c}_{12}+a_{13}{c}_{13}+a_{14}{c}_{14}$ equals

• A. $0$
• B. $-1$
• C. $1$
• D. $\left|A\right|$

Answer 1

Answer: (D)

$\left|A\right|$

$A=\left[a_{ij}\right]4\times 4$

$c_{ij}\to$co factor

$A=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$ co factor $=$ Minor $\times \begin{array}{ccc}+& -& +\\ -& +& -\\ +& -& +\end{array}$

Co factor matrix$=\left[\begin{array}{ccc}{a}_{22}{a}_{33}-a_{23}{a}_{32}& -a_{21}{a}_{33}+a_{23}{a}_{31}& a_{21}{a}_{32}-a_{22}{a}_{31}\\ {-a}_{12}{a}_{33}+a_{13}{a}_{32}& a_{11}{a}_{33}-a_{13}{a}_{31}& -a_{11}{a}_{32}+a_{12}{a}_{31}\\ a_{12}{a}_{23}-a_{13}{a}_{22}& -a_{11}{a}_{23}+a_{13}{a}_{21}& a_{11}{a}_{22}-a_{12}{a}_{21}\end{array}\right]$

$c_{11}=a_{22}{a}_{33}-a_{23}{a}_{32}$

$c_{12}=a_{23}{a}_{31}-a_{21}{a}_{33}$

$c_{13}=a_{21}{a}_{32}-a_{22}{a}_{31}$

$a_{11}{c}_{11}+a_{12}{c}_{12}+a_{13}{c}_{13}$

$=\left|A\right|$

Parallelly For $4\times 4$ matrix

Also

$a_{11}{c}_{11}+a_{12}{c}_{12}+a_{13}{c}_{13}+a_{14}{c}_{14}=\left|A\right|$

Option D