The inverse of a Square Matrix

2.1 Day 1(Out of Pre-Calc

book 8.3)

We are reloading for another chapter

Definition of Inverse

• The inverse of a matrix is the matrix that you multiply times it to obtain I

• A*A-1 = I = A-1ANote: matrix multiplication is NOT generally commutative but it is in the case of square invertible matrices. Our books defines inverses for square matrices some books include inverses for rectangular matrices (would these be commutative?)

Example 1 (pre-Calc)

Example 1 solution

To solve Ax=b

One method would be to multiply both sides of the equation by A inverse. (the matrix that when multiplied times A yields the identity matrix.)

1 3

2 6

How to find an inverse

Inverse in terms of Determinants (p. 586)

• For a 2x2 matrix we can find the inverse of a matrix using determinants

Example 3Find the inverse of the matrix

Example 3 solution

What causes an inverse to fail to exist?

• One column is a multiple of another column

• If the determinant = 0

• If there are any non zero vectors x that are solutions to the equation Ax= 0

• If any row is a multiple of another row

Example 5

Solve the system of equations using the inverse of a matrix. Use technology to check your answer.

Example 5 Solution

First, write the system in the form Ax=b

Next, find A inverse

Finally, multiple both sides of the equation by A inverse

When would it be beneficial to use this method to solve systems of equations?

Example 5Solve the system of equations using the

inverse of a matrix. Use technology to check your answer.

To check the final answer use rref as shown in a previous lecture.To find the inverse of a matrix on a TI 89 Calculator. Enter the matrix and raise it to the -1 power

Homework p. 588 (pre-Calc)1-63 every other odd

1 0 0

0 1 0

0 0 1[ ]

Superman

=

Clark Kent

By: Mr. Whitehead