2.5
Determinants 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
a b
c d
 ad  bc
 5  7 -40 – -77
 -40 + 77
11 8
37
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 a
Third-Order Determinant
2
6
3 4
5
5 7 2
9
1 9 8
7
6 7
6 5
3
4
8
1 8
1 9
2(40 – 63) – 3(48 – -7) + 4(54 – -5)
2(-23) – 3(55) + 4(59)
-46 – 165+ 236
25
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.
2  1
1  3


= (2 ∙ -3) – (-1 ∙ 1)
= (-6) – (-1)
= -5
• Step 3 – Is the determinant zero?
• Step 4 – Fill in the formula.
2  1
Q

1  3
1  3 1 
Q  

5   1 2
1
Find the inverse of the matrix, if possible. If not
possible, explain why.
 4 6
6 9 


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