Chapter 8.3, Problem 21E

To determine

Tofind:the inverse of the matrix if it’s exists.

Answer to Problem 21E

The inverse of given matrix doesn’t exist.

Explanation of Solution

Given:

  $\text{A=}\left[\begin{array}{ccc}\text{-5}& \text{0}& \text{0}\\ \text{2}& \text{0}& \text{0}\\ \text{-1}& \text{5}& \text{7}\end{array}\right]$

Concept used:

Steps to solve the inverse of matrix by

Step1: calculating the matrix of minors.

Step2:turn that into the matrix of cofactors.

Step3: then ad jugate by transposing the cofactor.

Step4: last multiply by the $\frac{1}{\text{determinant}}$ .

Formula for inverse of matrix A is:

  ${\text{A}}^{\text{-1}}\text{=}\frac{\text{adjA}}{\left|\text{A}\right|}$

Calculation:

Here $\text{A=}\left[\begin{array}{ccc}\text{-5}& \text{0}& \text{0}\\ \text{2}& \text{0}& \text{0}\\ \text{-1}& \text{5}& \text{7}\end{array}\right]$

Cofactor of A: $\left[\begin{array}{ccc}0& \text{-}\left(14-0\right)& 10-0\\ -\left(0\right)& -35-0& -\left(-25-0\right)\\ 0& -\left(0\right)& 0\end{array}\right]$

Cofactor of A: $\left[\begin{array}{ccc}0& -14& 10\\ 0& -35& 25\\ 0& 0& 0\end{array}\right]$

Ad jugate of A: ${\left[\begin{array}{ccc}0& -14& 10\\ 0& -35& 25\\ 0& 0& 0\end{array}\right]}^{\text{T}}=\left[\begin{array}{ccc}0& 0& 0\\ -14& -35& 0\\ 10& 25& 0\end{array}\right]$

  ${\text{A}}^{\text{-1}}\text{=}\frac{\text{adjA}}{\left|\text{A}\right|}$

  $\begin{array}{l}\left|\text{A}\right|=-5\left(0-0\right)-0\left(14-0\right)+0\left(10-0\right)\\ \Rightarrow 0\\ \left|\text{A}\right|\text{=0}\end{array}$

  $$

The inverse of A is

  ${\text{A}}^{\text{-1}}\text{=}\frac{\left[\begin{array}{ccc}0& 0& 0\\ -14& -35& 0\\ 10& 25& 0\end{array}\right]}{0}$ .

Since the denominator is 0 so, it’s impossible to solved the matrix further

Hence, the inverse of given matrix doesn’t exist.