Question

If matrix $ A = \begin{bmatrix}1&0&-1\\ 3&4&5\\ 0&6&7\end{bmatrix}$ and its inverse is denoted by $A^{-1} = \begin{vmatrix}a_{11}&a_{12}&a_{13}\\ a_{21} &a_{22}&a_{23}\\ a_{31} &a_{32}&a_{33}\end{vmatrix} $ then the value of $a_{23}$ is equal to :

• A. 21/20
• B. 1/5
• C. -2/5
• D. 2/5

Answer 1

Answer: (C)

Given : $A = \begin{bmatrix}1&0&-1\\ 3&4&5\\ 0 &6&7\end{bmatrix} $
$\therefore \ | A | = 1 (-2) - 1 (18) = - 20 $
we know that $A^{-1} = \frac{Adj.A}{|A|}$
The element $a_{23}$ will be $\frac{A_{32}}{|A|}$, because Adj.
A is the transpose of the respective cofactors founded.
Now, $A_{32} = 5 - (-3) = 8$
Thus $a_{23} = \frac{8}{-20} = \frac{-2}{5}$