In today’s class we recapped from the investigation yesterday. Yesterday’s investigation was meant to be an introduction to the idea of an *inverse of a matrix*.

**Inverse of a Matrix**

- For a given matrix [A], the inverse of the matrix would be noted as [A]-1
- Only square matrices have inverses
- The inverse of a matrix can be found by hand or by calculator
- When a matrix [A] is multiplied by its inverse [A]-1, the result is an
**identity matrix I**

[A][A]-1=I
- Identity matrices take on the form:

[1 0

0 1]

Where there is a diagonal of “1”s
- The identity matrix is like the number 1 in matrices
- The idenity matrix has the property AI = IA= A ; if we multiply an identity matrix with matrix A, the result will be the same as the original matrix A

**Why Would We Need An Inverse?**

- If we consider the equation 3x=12, we need to isolate x for our solution
- we divide the left side by 3 which is the equivalent of multiplying by 1/3
- 1/3 is the inverse of 3
- we similarly need inverses of matrices to isolate our variables; we multiply both sides by the inverse of the coefficient matrix to isolate our x, y, z…

Note: when we multiply matrices, remember that the first matrix must have as many columns as the second matrix has in rows.

**Solving Matrices By Hand**

1) First find what we call the **determinant.** For the matrix:

A = [a b

c d]

The determinant is det(A)= ad-bc

If det(A)= 0, there is no solution or an infinite number of solutions

If det(A) does not equal 0, there is a unique solution

2) If there is a determinant,

A-1 = (1/det(A)) [d -b

-c d]

“1 divided by the determinant multipled by the rearranged matrix”

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