Question 8.4.2: Finding the Multiplicative Inverse of a Matrix Find the mult...

Let us denote the multiplicative inverse by

A^{-1} =\begin{bmatrix}w &x \\ y & z \end{bmatrix}.

Because A is a 2 × 2 matrix, we use the equation AA^{-1} = I_2 to find values for w, x, y, and z.

\begin{bmatrix}2w + y &2x + z \\ 5w + 3y & 5x + 3z\end{bmatrix} = \begin{bmatrix}1 &0 \\ 0 & 1 \end{bmatrix}                  Use row-by-column matrix                                                                                         multiplication on the left side of                                                                                     \begin{bmatrix}2&1 \\ 5 & 3\end{bmatrix}\begin{bmatrix}w &x \\ y & z \end{bmatrix}=\begin{bmatrix}1 &0 \\ 0 & 1 \end{bmatrix}.

We now equate corresponding elements to obtain the following two systems of linear equations:

\begin{cases}2w + y = 1\\ 5w + 3y = 0\end{cases}            and          \begin{cases}2x + z = 0\\ 5x + 3z = 1.\end{cases}

Each of these systems can be solved using the addition method.

w = 3

Use back-substitution.                y = -5

x = -1

Use back-substitution.             z = 2

Using these values, we have

A^{-1}=\begin{bmatrix} w &x \\ y & z \end{bmatrix}=\begin{bmatrix}3 &-1 \\ -5 & 2 \end{bmatrix}.