Chapter 7, Problem 106CR

To determine

To find: the inverse of the matrix (if it exists) using the matrix capabilities of a graphing utility.

Answer to Problem 106CR

  $A^{-1}=\left[\begin{array}{cc}1.5& -2\\ -3.5& 5\end{array}\right]$ .

Explanation of Solution

Given information: Given matrix is $\left[\begin{array}{cc}10& 4\\ 7& 3\end{array}\right].$

Formula used.

If A is the given Matrix then,

  $A^{-1}=\frac{Adj(A)}{\left|A\right|}.$

Calculation:

Given matrix for which inverse is to be found, is shown below.

  $A=\left[\begin{array}{cc}10& 4\\ 7& 3\end{array}\right].$

Cofactor matrix for the given matrix is shown below.

  $\left[\begin{array}{cc}3& -7\\ -4& 10\end{array}\right]$ .

Adjoint matrix is the transpose of the cofactor matrix is shown below.

  $Adj(A)=\left[\begin{array}{cc}3& -4\\ -7& 10\end{array}\right]$ .

Determinant of the given matrix is 2,

  $\left|A\right|=\left[\begin{array}{cc}10& 4\\ 7& 3\end{array}\right]=30-28=2.$

Hence the inverse of the given matrix is,

  $\begin{array}{l}{A}^{-1}=\frac{1}{2}\times \left[\begin{array}{cc}3& -4\\ -7& 10\end{array}\right].\\ A^{-1}=\left[\begin{array}{cc}1.5& -2\\ -3.5& 5\end{array}\right].\end{array}$