Solve the matrix equation:

|4 -5||m|=|32|
|1 2||n|=|-5|

I don't get it

If I read your matrices correctly, multiplying them would result in the two linear equations:

4m - 5n = 32 and
m + 2n = -5

I am sure you can solve those

thats not how they solve it in my book. the steps are to find the inverse of the coefficient matrix then multiply each side of the matrix by the inverse matrix

A= |4 -5|

|1 2|

X= |m|
|n|

B= |32|
|-5|

To solve the matrix equation, you need to find the values of variables m and n that satisfy the equation. Here's how you can do it step by step:

Step 1: Set up the equation
Start by writing the matrix equation in the form A * X = B, where A is the coefficient matrix, X is the column matrix containing variables m and n, and B is the column matrix on the right-hand side.

|4 -5| |m| = |32|
|1 2| |n| = |-5|

Step 2: Inverse of matrix A
Find the inverse of matrix A, denoted as A^-1. The inverse allows us to solve for X by multiplying both sides of the equation by A^-1.

Step 3: Multiply both sides by A^-1
Multiply both sides of the equation by A^-1:

A^-1 * A * X = A^-1 * B

The A^-1 * A product will result in the identity matrix I, so the equation simplifies to:

I * X = A^-1 * B

Step 4: Calculate A^-1
Calculate the inverse of matrix A. In this case, A is a 2x2 matrix, which makes it relatively simple to find the inverse.

To find the inverse of a 2x2 matrix, swap the positions of the elements on the main diagonal (top-left to bottom-right), and change the signs of the elements on the other diagonal (bottom-left to top-right). Then, divide each element of the resulting matrix by the determinant of A (ad - bc).

A = |4 -5|
|1 2|

The determinant of A (ad - bc) is calculated as follows:
(4 * 2) - (-5 * 1) = 8 + 5 = 13

Swap positions and change signs:
A^-1 = |2 5|
|-1 -4|

Divide each element by the determinant value (13):
A^-1 = |2/13 5/13|
|-1/13 -4/13|

Step 5: Multiply A^-1 by B
Multiply the inverse of matrix A, A^-1, by matrix B:

A^-1 * B = |2/13 5/13| * |32| = |(2/13)*32 + (5/13)*(-5)| = |64/13 - 25/13|

Step 6: Final solution
The resulting matrix on the right side of the equation is the solution vector X. In this case, X is a column matrix containing the values of m and n.

X = |64/13|
|-25/13|

Therefore, the solution to the given matrix equation is m = 64/13 and n = -25/13.