Chapter A.2, Problem 3PP

To determine

Find the solution to the given simultaneous equations by matrix inversion method.

Answer to Problem 3PP

The solutions to the given simultaneous equations by matrix inversion method are $\underset{\_}{y_1=3\text{and}{y}_2=2}$.

Explanation of Solution

Given data:

$2y_1-y_2=4$ (1)

$y_1+3y_2=9$ (2)

Formula used:

Consider the general expression to determine the matrix inversion.

$X=A^{-1}B$ (3)

Here,

$A^{-1}=\frac{1}{\left|A\right|}\left[\begin{array}{cc}d& -b\\ -c& a\end{array}\right]$

Calculation:

Evaluate given set of Equation (1) and Equation (2) in matrix form as follows.

$\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]=\left[\begin{array}{c}4\\ 9\end{array}\right]$

Here,

$\begin{array}{l}A=\left[\begin{array}{cc}2& -1\\ 1& 3\end{array}\right]\\ X=\left[\begin{array}{c}{y}_1\\ y_2\end{array}\right]\\ B=\left[\begin{array}{c}4\\ 9\end{array}\right]\end{array}$

Find determinants of matrix A.

$\begin{array}{c}\left|A\right|=\left|\begin{array}{cc}2& -1\\ 1& 3\end{array}\right|\\ =\left[\left(2\right)\left(3\right)\right]-\left[\left(-1\right)\left(1\right)\right]\\ =6+1\\ =7\end{array}$

Find inverse of A.

$A^{-1}=\frac{1}{7}\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$

Substitute $\frac{1}{7}\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ for $A^{-1}$ and $\left[\begin{array}{c}4\\ 9\end{array}\right]$ for $B$ in equation (3) as follows.

$\begin{array}{c}X=\frac{1}{7}\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]\left[\begin{array}{c}4\\ 9\end{array}\right]\\ =\frac{1}{7}\left[\begin{array}{c}21\\ 14\end{array}\right]\\ =\left[\begin{array}{c}\frac{21}{7}\\ \frac{14}{7}\end{array}\right]\\ =\left[\begin{array}{c}3\\ 2\end{array}\right]\end{array}$

Conclusion:

Thus, the solutions to the given simultaneous equations by matrix inversion method are $\underset{\_}{y_1=3\text{and}{y}_2=2}$.