Answer to Question #265798 in Linear Algebra for Good heart

Question

Solve the following pair of linear equation by matrix method 2p +q =5

5p +3=11

Answer 1

The coefficient matrix is:

$$A=\begin{pmatrix} 2 & 1 \\ 5 & 0 \end{pmatrix}$$

The variable matrix is:

$$X=\begin{pmatrix} p \\ q \end{pmatrix}$$

The constant matrix is:

$$B=\begin{pmatrix} 5 \\ 8 \end{pmatrix}$$

Thus, to solve a system $AX=B,$ for $X,$ multiply both sides by the inverse of $A$

$$A^{-1}AX=A^{-1}B$$

and we shall obtain the solution:

$$X=A^{-1}B$$

Provided the inverse $A^{-1}$ exists, this formula will solve the system.

$$\det A=|A|=\begin{vmatrix} 2 & 1 \\ 5 & 0 \end{vmatrix}=2(0)-1(5)=-5\not=0$$

The inverse $A^{-1}$ exists.

$$A^{-1}=\dfrac{1}{-5}\begin{pmatrix} 0 & -1 \\ -5 & 2 \end{pmatrix}=\begin{pmatrix} 0 & 1/5 \\ 1 & -2/5 \end{pmatrix}$$

$$X=A^{-1}B=\begin{pmatrix} 0 & 1/5 \\ 1 & -2/5 \end{pmatrix}\begin{pmatrix} 5 \\ 8 \end{pmatrix}$$

$$=\begin{pmatrix} 0(5)+(1/5)(8) \\ 1(5)+(-2/5)(8) \end{pmatrix}=\begin{pmatrix} 8/5 \\ 9/5 \end{pmatrix}$$

$$\begin{pmatrix} p \\ q \end{pmatrix}=\begin{pmatrix} 8/5 \\ 9/5 \end{pmatrix}$$

$$p=8/5, q=9/5$$