Question

solve the set of equations by the method of finding the inverse of the coefficient matrix.

$$\begin{gathered} \{\begin{array}{l}2x+3y=-1 \\ 5x+4y=8\\ \end{array} \end{gathered}$$

Short answer

The solution of the given set of equations is x=4 and y=-3 .

Step-by-step solution

Definition of inverse of a matrix:

The inverse of a matrix M is defined as the matrix ${\mathit{M}}^{\mathbf{-}\mathbf{1}}$ such that $\mathit{M}{\mathit{M}}^{\mathbf{-}\mathbf{1}}$ and ${\mathit{M}}^{\mathbf{-}\mathbf{1}}\mathit{M}$ are both equal to a unit matrix I .

 Find the inverse of the matrix:

The given set of equations are

$$\begin{gathered} \left\{\begin{array}{l}2x+3y=-1 \\ 5x+4y=8\\ \end{array}\right. \end{gathered}$$

The solution is to be found by the method of finding the inverse of the coefficient matrix. Here,

$$\begin{gathered} M=\left(23 \\ 54\right) \end{gathered}$$

So,

$$\begin{gathered} C=\left(C_{11}{C}_{12} \\ {C}_{21}{C}_{22}\right) \\ =\left(4-5 \\ -32\right) \end{gathered}$$

Find inverse of matrix M .

$$\begin{gathered} M^{-1}=\frac{1}{detM}{C}^T \\ =\frac{1}{\left(4\times 2-3\times 5\right)}\left(4-3 \\ -52\right) \\ =\frac{1}{7}\left(-43 \\ 5-2\right) \end{gathered}$$

 Find the solution of equations:

Use equation $M^{-1}k=r$to solve the equations, where is the matrix of coefficient and is the unknown vector. Here, $$\begin{gathered} k=\left(-1 \\ 8\right) \end{gathered}$$ and $$\begin{gathered} r=\left(x \\ y\right) \end{gathered}$$ .

$$\begin{gathered} \frac{1}{7}\left(-43 \\ 5-2\right)\left(-1 \\ 8\right)=\left(x \\ y\right) \\ \frac{1}{7}\left(28 \\ -21\right)=\left(x \\ y\right) \\ \left(4 \\ -3\right)=\left(x \\ y\right) \end{gathered}$$

Therefore, x=4 and y=-3 .