Question

If $A=\left[\begin{array}{ccc}1& 1& 0\\ 2& 1& 5\\ 1& 2& 1\end{array}\right]$, then $a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}=$.......

• A. $1$
• B. $0$
• C. $-1$
• D. $2$

Answer 1

Answer: (B)

$0$

Here, $a_{ij}$ stands for element of $A$ with $i^{th}$ row and $j^{th}$ column. Also, $A_{ij}$ stands for co-factor of $a_{ij}$ of matrix $A$

So, we get $a_{11}=1,a_{12}=1$ and $a_{13}=0$

To find the co-factor

$A_{21}=(-1{)}^{2+1}\left|\begin{array}{cc}1& 0\\ 2& 1\end{array}\right|=(-1\left)\right(1-0)=-1$

$A_{22}=(-1{)}^{2+2}\left|\begin{array}{cc}1& 0\\ 1& 1\end{array}\right|=(-1{)}^4(1-0)=1$

$A_{23}=(-1{)}^{2+3}\left|\begin{array}{cc}1& 1\\ 1& 2\end{array}\right|=(-1{)}^5(2-1)=-1$

$\therefore a_{11}{A}_{21}+a_{12}{A}_{22}+a_{13}{A}_{23}=1(-1)+1\left(1\right)+0(-1)=0$