Chapter 2Chapter 2Systems of Linear Equations Systems of Linear Equations

and Matricesand Matrices

Section 2.5Section 2.5

Matrix InversesMatrix Inverses

What is a Matrix Inverse?What is a Matrix Inverse?

The inverse of a matrix is The inverse of a matrix is comparablecomparable to the reciprocal of a real number.to the reciprocal of a real number.

The product of a matrix and its identity The product of a matrix and its identity matrix is always the matrix itself. matrix is always the matrix itself. In other words, multiplying a matrix by In other words, multiplying a matrix by its identity matrix is like multiplying a its identity matrix is like multiplying a number by 1.number by 1.

Multiplicative IdentityMultiplicative Identity The real number 1 is the multiplicative The real number 1 is the multiplicative

identity for real numbers:identity for real numbers:for any real number for any real number aa, we have , we have aa • 1 = 1 • • 1 = 1 • aa = = aa

In this section, we define a In this section, we define a multiplicative multiplicative identity matrix identity matrix II that has properties similar that has properties similar to those of the number 1.to those of the number 1.

We use the definition of this matrix We use the definition of this matrix II to to find the multiplicative inverse of any find the multiplicative inverse of any square matrix that has an inverse.square matrix that has an inverse.

Identity MatrixIdentity Matrix

If If II is to be the identity matrix, both is to be the identity matrix, both of the products of the products AI AI andand IA IA must equal must equal A.A.

The identity matrix only exists for The identity matrix only exists for square matrices.square matrices.

Examples of Identity MatricesExamples of Identity Matrices

Determining if Matrices are Determining if Matrices are Inverses of Each OtherInverses of Each Other

Recall that a number multiplied by its Recall that a number multiplied by its multiplicative inverse yields a product of 1.multiplicative inverse yields a product of 1.

Similarly, the product of matrix Similarly, the product of matrix AA and its and its multiplicative inverse matrix multiplicative inverse matrix AA (read “A- (read “A-inverse”) is inverse”) is II, the identity matrix., the identity matrix.

So, to prove that two matrices are inverses So, to prove that two matrices are inverses of each other, show that their product, of each other, show that their product, regardless of the order they’re multiplied, is regardless of the order they’re multiplied, is always the identity matrix.always the identity matrix.

1

Example 1Example 1 Prove or disprove that the matrices below are Prove or disprove that the matrices below are

inverses of each other.inverses of each other.

a.) a.)

b.)b.)

c.)c.)

5 7 3 7

2 3 2 5and

1 2 5 2

3 5 3 1and

0 1 0 1 0 1

0 0 2 1 0 0

1 1 0 0 1 0

and

Finding the Inverse of a MatrixFinding the Inverse of a Matrix

Row Operations on MatricesRow Operations on Matrices

Example 2Example 2

Find the inverse, if it exists, for each Find the inverse, if it exists, for each matrix.matrix.

a.)a.) b.)b.)

c.)c.)

1 2

3 4

5 10

3 6

1 2

2 1

Shortcut for Finding the Inverse of Shortcut for Finding the Inverse of a 2 x 2 Matrixa 2 x 2 Matrix

If a matrix is of the formIf a matrix is of the form

then the inverse can be found by then the inverse can be found by calculating:calculating:

Note: ad – bc ≠ 0.Note: ad – bc ≠ 0.

a b

c d

1 d b

c aad bc

Example 3Example 3

Find the inverse of the matrix below using Find the inverse of the matrix below using the shortcut method.the shortcut method.

4 2

5 3

Solution to Example 3Solution to Example 3

1 d b

c aad bc

4 2

5 3

To find the inverse of the matrix

use the formula and simplify.

3 21

5 44(3) 2(5)

Solution to Example 3 (continued)Solution to Example 3 (continued)

1

3 21

5 44(3) 2(5)

3 21

5 42

31

25

22

A