Chapter 7, Problem 104CR

To determine

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

Answer to Problem 104CR

  $A^{-1}\text{does}\text{not}\text{exist}.$

Explanation of Solution

Given information: Given matrix is $\left[\begin{array}{ccc}1& -1& -2\\ 0& 3& -2\\ 1& 2& -4\end{array}\right].$

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.

  $A=\left[\begin{array}{ccc}1& -1& -2\\ 0& 3& -2\\ 1& 2& -4\end{array}\right].$

Cofactor matrix for the given matrix is shown below.

  $\left[\begin{array}{ccc}-8& -2& -3\\ -8& -2& -3\\ 8& 2& 3\end{array}\right]$ .

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

  $Adj(A)=\left[\begin{array}{ccc}-8& -8& 8\\ -2& -2& 2\\ -3& -3& 3\end{array}\right].$ .

Determinant of the given matrix is 0.

  $\left|A\right|=\left[\begin{array}{ccc}1& -1& -2\\ 0& 3& -2\\ 1& 2& -4\end{array}\right]=0.$

  $A^{-1}=(\frac{1}{0})\times \left[\begin{array}{ccc}-8& -8& 8\\ -2& -2& 2\\ -3& -3& 3\end{array}\right]=\text{Does}\text{not}\text{exist}.$

Hence the inverse of the given matrix does not exist because determinant of the matrix is zero.

  $A^{-1}\text{does}\text{not}\text{exist}.$