Why are inverse matrices important to solving equations ?

Question

Algebra II

Answer 1

Because when you multiply a matrix by its inverse
you get the identity matrix.

Answer 2

if we have a matrix, let's call it A, then we multiply it by its inverse, which we will call A^-1, we get something called an identity matrix, which is a matrix with all 1's in the diagonal and 0 everywhere else.

So we have a matrix A times the column matrix, which are its variables.... like this

| 3 4 | . | x |......| 1 |
| 7 6 | * | y | = | 2 |

this is the same as having the system of equations

3x + 4y = 1
7x + 6y = 2

Now i can use the property of A*A^-1 = I

| 1 0 |*| x |.....-1/6 * | 6 -4 | * | 1 |
| 0 1 | .| y | = ..........| -7 3 | | 2 |

Where i have multiplied the equation by A^-1. Now I have on the left,

x = some number
y = some number

and we have solved for x and y. Ignore the periods everywhere... they were just to make everything line up.

Answer 3

Linear equations can be written in the form Ax = B, where A and B are matrices and x is a column vector (the variables). You can then solve the equation by multiplying on the left by the inverse of A, to get x = A^-1 * B.
So thats why they are important.