Chapter 4.6, Problem 32PPS

To determine

To solve: the matrix equations.

Answer to Problem 32PPS

No, solution.

Explanation of Solution

Given information:

  $\begin{array}{l}4x+2y=6\\ 6x-3y=9\end{array}$ .

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,

  $\begin{array}{l}4x+2y=6\\ 6x-3y=9\end{array}$

The matrix equation is

  $\left[\begin{array}{cc}4& 2\\ 6& -3\end{array}\right].\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}6\\ 9\end{array}\right]$ .

Step 1.

  $=A^{-1}=\frac{1}{(4)(-3)-(2)(6)}\left[\begin{array}{cc}-3& -2\\ -6& 4\end{array}\right]$

  $=\frac{1}{-12-12}\left[\begin{array}{cc}-3& -2\\ -6& 4\end{array}\right]$

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

Therefore, the system of equation has no solution.