Chapter 4.6, Problem 27PPS

To determine

To solve: the matrix equations.

Answer to Problem 27PPS

No, solution for the system .

Explanation of Solution

Given information:

  $\begin{array}{l}-x+y=4\\ -x+y=-4\end{array}$ .

Formula used:

Inverse of a $2\times 2$ matrix $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$ is $A^{-1}=\frac{1}{ad-bc}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$ is, where $ad-bc\ne 0$ .

The value of a second order determinant is the difference of the products of the products of the two diagonals.

Steps to solve the equations are:-

Step1. Find the inverse of the coefficient matrix.

Step 2. Multiple each side of the matrix equation by the inverse matrix.

Calculation:

Consider,

  $\begin{array}{l}-x+y=4\\ -x+y=-4\end{array}$

The matrix equation is

  $\left[\begin{array}{cc}-1& 1\\ -1& 1\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}4\\ -4\end{array}\right]$ .

Step 1.

  $=A^{-1}=\frac{1}{(-1)(1)-(1)(-1)}\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$

  $=\frac{1}{-1+1}\left[\begin{array}{cc}1& -1\\ 1& -1\end{array}\right]$

Since the determinant is zero. Therefore, inverse does not exist.

Thus, there is no solution for the system..