Chapter 7, Problem 10RE

Solution

To calculate: The multiplication of the matrix and explain that the matrices are inverse.

They are inverses.

Given Information:

The matrix are defined as,

  $A=\left[\begin{array}{ccc}-1& 1& 1\\ 2& 1& 0\\ -1& 0& 2\end{array}\right],B=\left[\begin{array}{ccc}-0.4& 0.4& 0.2\\ 0.8& 0.2& -0.4\\ -0.2& 0.2& 0.6\end{array}\right]$

Formula Used:

Two matrices are said to be inverse if $AB=BA=I_n$ .

Where, the matrix $I_n$ is the identity matrix.

Calculation:

Consider the given matrices,

  $A=\left[\begin{array}{ccc}-1& 1& 1\\ 2& 1& 0\\ -1& 0& 2\end{array}\right],B=\left[\begin{array}{ccc}-0.4& 0.4& 0.2\\ 0.8& 0.2& -0.4\\ -0.2& 0.2& 0.6\end{array}\right]$

Find the multiplication.

  $\begin{array}{c}AB=\left[\begin{array}{ccc}-1& 1& 1\\ 2& 1& 0\\ -1& 0& 2\end{array}\right]\cdot \left[\begin{array}{ccc}-0.4& 0.4& 0.2\\ 0.8& 0.2& -0.4\\ -0.2& 0.2& 0.6\end{array}\right]\\ =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\text{From calculator}\right]\end{array}$

And,

  $\begin{array}{c}BA=\left[\begin{array}{ccc}-0.4& 0.4& 0.2\\ 0.8& 0.2& -0.4\\ -0.2& 0.2& 0.6\end{array}\right]\cdot \left[\begin{array}{ccc}-1& 1& 1\\ 2& 1& 0\\ -1& 0& 2\end{array}\right]\\ =\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\text{From calculator}\right]\end{array}$

As the obtained matrix is the identity matrix of order 3 cross 3.

Hence, the given matrices are inverse.