Chapter 8.4, Problem 128E

To determine

To calculate: the inverse of the matrix.

Answer to Problem 128E

Inverse of the matrix $\left[\begin{array}{cc}11& -12\\ 2& -2\end{array}\right]$ is $\underset{\_}{\left[\begin{array}{cc}-1& 6\\ -1& \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]}$ .

Explanation of Solution

Given information:

Matrix for which inverse needs to be calculated is as follows.

  $\left[\begin{array}{cc}11& -12\\ 2& -2\end{array}\right]$

Formula used:

For matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$

  $A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Where $\left|A\right|=ad-bc$ .

Calculation:

Matrix for which inverse needs to be calculated is as follows.

  $A=\left[\begin{array}{cc}11& -12\\ 2& -2\end{array}\right]$

For matrix $A$ , the determinant is

  $\left|A\right|=ad-bc=(11)(-2)-(-12)(2)$

  $=-22+24$

  $=2$

Interchange the diagonal elements and the sign accordingly to the formula and apply formula of $A^{-1}$ we get,

  $A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}-2& 12\\ -2& 11\end{array}\right]$

  $=\frac{1}{2}\left[\begin{array}{cc}-2& 12\\ -2& 11\end{array}\right]$

  $=\left[\begin{array}{cc}-\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$12$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\\ -\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$2$}\right.& \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]$

  $=\left[\begin{array}{cc}-1& 6\\ -1& \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]$

Hence inverse of the matrix $\left[\begin{array}{cc}11& -12\\ 2& -2\end{array}\right]$ is $\underset{\_}{\left[\begin{array}{cc}-1& 6\\ -1& \raisebox{1ex}{$11$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\end{array}\right]}$ .