Chapter SH, Problem 3.26EP

To determine

To solve: the solution of the given system of equation by using matrix.

Answer to Problem 3.26EP

No solution.

Explanation of Solution

Given:

The given system of equation is $\begin{array}{l}2x+3y=6\\ 6x+9y=-10\end{array}$ .

Concept used:

By using matrix, the formula of system of linear equation is given below:

  $\begin{array}{l}ifA=\left[\begin{array}{l}ab\\ cd\end{array}\right]\\ then{A}^{-1}=\frac{1}{ad-bc}\left[\begin{array}{l}d-b\\ -ca\end{array}\right]\end{array}$

Calculation:

The givensystem of equation is $\begin{array}{l}2x+3y=6\\ 6x+9y=-10\end{array}$ .

Therefore, the matrix form of equation is $\left[\begin{array}{l}23\\ 69\end{array}\right]$

Now, to find the inverse of A so that to find the solution of the system of equation:

  $\begin{array}{l}{A}^{-1}=\frac{1}{2\left(9\right)-\left(3\right)\left(6\right)}\left[\begin{array}{l}9-3\\ -62\end{array}\right]\\ =\frac{1}{18-18}\left[\begin{array}{l}9-3\\ -62\end{array}\right]=\frac{1}{0}\left[\begin{array}{l}9-3\\ -62\end{array}\right]\end{array}$

Here the determinant of adjoint matrix is 0, therefore, the are linearly dependent.

Hence, from matrix rule, the solution of the given system of equation has no solution.