Question

Solve the following system of equations using an inverse matrix. You must alsoindicate the inverse matrix, A-1, that was used to solve the system. You mayoptionally write the inverse matrix with a scalar coefficient.5x-3y = 87x-5y=4

Answer 1

Step 1:

Write the two equation

5x - 3y = 8

7x - 5y = 4

Step 2

Write in matrix form

$\begin{gathered} \begin{bmatrix}{5} & {-3} \\ {7} & {-5}\end{bmatrix}\begin{bmatrix}{x} & {} \\ {y} & {}\end{bmatrix}=\begin{bmatrix}{8} & {} \\ {4} & {}\end{bmatrix} \\ Inverse\text{ A}^{-1}\text{ = }\frac{Adjoint}{|A|} \\ A^{-1}\text{ = }\begin{bmatrix}{5} & {-3} \\ {7} & {-5}\end{bmatrix}^{-1} \\ \begin{bmatrix}{x} & {} \\ {y} & {}\end{bmatrix}=\text{ }\begin{bmatrix}{5} & {-3} \\ {7} & {-5}\end{bmatrix}^{-1}\begin{bmatrix}{8} & {} \\ {4} & {}\end{bmatrix} \end{gathered}$

Step 3:

$\begin{gathered} \begin{bmatrix}{5} & {-3} \\ {7} & -{5}\end{bmatrix} \\ Determinant\text{ = -25+21 = -4} \\ Cofactor\text{ = }\begin{bmatrix}{-5} & {-7} \\ {3} & {5}\end{bmatrix} \\ Adjoint\text{ = }\begin{bmatrix}{-5} & {3} \\ {-7} & {5}\end{bmatrix} \\ A^{-1}\text{ = }\frac{1}{-4}\begin{bmatrix}{-5} & {3} \\ {-7} & {5}\end{bmatrix} \end{gathered}$

Step 4:

$\begin{gathered} \begin{bmatrix}{x} & {} \\ {y} & {}\end{bmatrix}\text{ = }\frac{1}{-4}\begin{bmatrix}{-5} & {3} \\ {-7} & {5}\end{bmatrix}\begin{bmatrix}{8} & {} \\ {4} & {}\end{bmatrix} \\ =\text{ }\frac{1}{-4}\begin{bmatrix}{-40+12} & {} \\ {-56+20} & {}\end{bmatrix} \\ =\text{ }\frac{1}{-4}\begin{bmatrix}{-28} & {} \\ {-36} & {}\end{bmatrix} \\ \begin{bmatrix}{x} & {} \\ {y} & {}\end{bmatrix}=\text{ }\begin{bmatrix}{7} & {} \\ {9} & {}\end{bmatrix} \\ x\text{ = 7 , y = 9} \end{gathered}$

Final answer

$A^{-1}\text{ = }\frac{1}{-4}\begin{bmatrix}{-5} & {3} \\ {-7} & {5}\end{bmatrix}\text{ , x = 7 , y = 9}$