2.5Determinants and

Multiplicative Inverses of Matrices

Objectives:

Evaluate determinants.

Find inverses of matrices.

Solve systems of equations by using inverse matirces.

Determinants• Only square matrices have determinants.

Second-Order Determinant

bcaddc

ba

811

75 -40 – -77-40 + 7737

Third-Order Determinant• Determinants of 3x3 matrices are called third-order

determinants.

• One method of evaluating third-order determinants is called expansion by minors.

• The minor of an element is the determinant formed when the row and column containing the element are deleted.

Expansion of aThird-Order Determinant

891

756

432

91

564

81

763

89

752

2(40 – 63) – 3(48 – -7) + 4(54 – -5)2(-23) – 3(55) + 4(59)-46 – 165+ 23625

Third-Order Determinant• Another method for evaluating a third-order

determinant is using diagonals.

• In this method, you begin by writing the first two columns on the right side of the determinant.

Diagonals Method

Inverse of a 2x2 Matrix

• A matrix must be square to have an inverse.• A square matrix has an inverse if and only if

its determinant is not zero.

Inverse of a 2x2 Matrix

• The scalar is the reciprocal of the determinant.• a and d trade places• b and c change signs

• Step 1 – Is the matrix square?• Step 2 – Find the determinant of the matrix.

31

12 = (2 ∙ -3) – (-1 ∙ 1) = (-6) – (-1) = -5

• Step 3 – Is the determinant zero?• Step 4 – Fill in the formula.

31

12Q

21

13

5

11Q

Find the inverse of the matrix, if possible. If not possible, explain why.

96

64

Matrix Equation

Assignment

2.5 Practice Worksheet #1-8

Do #3 and 4 with minors and diagonals

2.5 pg 102 #18, 22, 23, 27, 28, 34, 38

Do #22 & 23 with both diagonals and minors