Question

Solve the matrix equation by using inverse matrices.

[2 -2] * [x] = [-18]
[-1 3] * [y] = [13 ]

Answer 1

x=-7, y=2

Explanation:

You are given the matrix equation

`\left[\begin{array}{cc}2&-2\\-1&3\end{array}\right] \cdot \left[\begin{array}{c}x\\y\end{array}\right] = \left[\begin{array}{c}-18\\13\end{array}\right]`

Find the inverse matrix for the matrix

`\left[\begin{array}{cc}2&-2\\-1&3\end{array}\right]`

1. The determinant is

`\left|\left[\begin{array}{cc}2&-2\\-1&3\end{array}\right]\right|=2\cdot 3-(-1)\cdot (-2)=6-2=4`

2.

`a_(11)=2 \Rightarrow A_(11)=3\\ \\a_(12)=-2 \Rightarrow A_(12)=-(-1)=1\\ \\a_(21)=-1 \Rightarrow A_(21)=-(-2)=2\\ \\a_(22)=3 \Rightarrow A_(22)=2`

3. Inverse matrix is

`(1)/(4)\left[\begin{array}{cc}3&1\\2&2\end{array}\right]^T=(1)/(4)\left[\begin{array}{cc}3&2\\1&2\end{array}\right]`

So, the solution of the equation is

`\left[\begin{array}{c}x\\y\end{array}\right]=(1)/(4)\left[\begin{array}{cc}3&2\\1&2\end{array}\right]\cdot \left[\begin{array}{c}-18\\13\end{array}\right]=\\ \\=(1)/(4)\left[\begin{array}{cc}3\cdot(-18)+2\cdot 13\\1\cdot (-18)+2\cdot 13\end{array}\right]=(1)/(4)\left[\begin{array}{c}-28\\8\end{array}\right]=\left[\begin{array}{c}-7\\2\end{array}\right]`