My book doesn't solve it like a linear equation, they solve it by using inverse matrices.
Solve the matrix equation:
|4 -5||m|=|32|
|1 2| |n|=|-5|
A= |4 -5|
|1 2|
X= |m|
|n|
B= |32|
|-5|
Step 1. Find the inverse of the coefficient matrix.
A^-1= 1/13|2 5|
|-1 4|
(I get confused in step 2)
Step 2. Multiply each side of the matrix by the inverse matrix
I am going to leave out the 1/13 on both sides of the matrix equation, so they line up nicely
│ 2 5││ 4-5││m│
│-1 4││ 1 2││n│ =
│ 2 5││32│
│-1 4││-5│
│13 0││m│
│0 13││n│ =
│ 39│
│-52│
so 13m=39 ----> m=3
and 13n=-52 --> n=-4
To solve the matrix equation using inverse matrices, you need to multiply each side of the equation by the inverse of the coefficient matrix. Let's continue with step 2.
The equation is:
A * X = B
We have already calculated the inverse of matrix A as A^-1:
A^-1 = 1/13 | 2 5 |
| -1 4 |
To find the solution matrix X, multiply both sides of the equation by A^-1:
A^-1 * (A * X) = A^-1 * B
The A matrix on the left side of the equation and the A^-1 matrix on the right side will cancel out, leaving us with:
(X) = A^-1 * B
Now, substitute the values we have:
X = A^-1 * B
X = 1/13 | 2 5 | * | 32 |
| -1 4 | * | -5 |
To calculate the values of the solution matrix X, perform the matrix multiplication:
X = 1/13 * (2*32 + 5*(-5)) = 1/13 * (64 - 25) = 1/13 * 39 = 3
Therefore, the value of m is 3 and the value of n is also 3, as they are both elements of the solution matrix X:
m = 3
n = 3
So the solution to the matrix equation is m = 3 and n = 3.