4.5 Matrices, Determinants, Inverseres

-Identity matrices-Inverse matrix (intro)-An application-Finding inverse matrices (by hand)-Finding inverse matrices (using calculator)

A review of the Identity

• For real numbers, what is the additive identity?

• Zero…. Why?

• Because for any real number b, 0 + b = b

• What is the multiplicative identity?

• 1 … Why?

• Because for any real number b, 1 * b = b

Identity Matrices

• The identity matrix is a square matrix (same # of rows and columns) that, when multiplied by another matrix, equals that same matrix

• If A is any n x n matrix and I is the n x n Identity matrix, then A * I = A and I*A = A

Examples

• The 2 x 2 Identity matrix is:

• The 3 x 3 Identity matrix is:

1 0

0 1

1 0 0

0 1 0

0 0 1

•Notice any pattern?

•Most of the elements are 0, except those in the diagonal from upper left to lower right, in which every element is 1!

Inverse review

• Recall that we defined the inverse of a real number b to be a real number a such that a and b combined to form the identity

• For example, 3 and -3 are additive inverses since 3 + -3 = 0, the additive identity

• Also, -2 and – ½ are multiplicative inverses since (-2) *(- ½ ) = 1, the multiplicative identity

Matrix Inverses

• Two n x n matrices are inverses of each other if their product is the identity

• Not all matrices have inverses (more on this later)

• Often we symbolize the inverse of a matrix by writing it with an exponent of (-1)

• For example, the inverse of matrix A is A-1

• A * A-1 = I, the identity matrix.. Also A-1 *A = I• To determine if 2 matrices are inverses, multiply

them and see if the result is the Identity matrix!

Determine whether X and Y are inverses.

Determine whether P and Q are inverses.

Determine whether each pair of matrices are inverses.

a.

b.

How do we find the inverse???

• 1st find the determinant• The determinant;

– determines whether the inverse of a matrix exists.– influences the elements the inverse contains

• For the matrix shown below, the determinant is equal to ad – bc

• In words, multiply the elements in each diagonal, then subtract the products!

• Order Matters.a b

c d

More about determinants

• If the determinant of a matrix equals zero, then the inverse of that matrix does not exist!

• Every square matrix has a determinant. • We will use DETERMINANTS and INVERSES to

solve matrix equations of the type AX = B.

Finding the inverse of a 2 x 2 matrix

• For the matrix:

• The inverse is found by calculating:

a b

c d

1 d b

c aad bc

In other words: -Switch the elements a and d -Reverse the signs of the elements b and c -Multiply by the fraction (1 / determinant)

Find the inverse of the matrix, if it exists.

Find the value of the determinant.

Since the determinant is not equal to 0, S –1 exists.

Find the inverse of each matrix, if it exists.

a.

b.

Answer: No inverse exists.

Answer:

Find the inverse of the matrix, if it exists.

Find the value of the determinant.

Answer: Since the determinant equals 0, T –1 does

not exist.

Definition of inverse

a = –1, b = 0,c = 8, d = –2

Answer: Simplify.

Check: