View
33
Download
0
Category
Tags:
Preview:
DESCRIPTION
Section 4.5. 2 x 2 Matrices, Determinants, and Inverses. Evaluating Determinants of 2 x 2 Matrices. Definition 1: A square matrix is a matrix with the same number of columns and rows. - PowerPoint PPT Presentation
Citation preview
Section 4.52 x 2 Matrices, Determinants, and Inverses
Definition 1: A square matrix is a matrix with the same number of columns and rows.
Definition 2: For an n x n square matrix, the multiplicative identity matrix is an n x n square matrix I, or In, with 1’s along the main diagonal and 0’s elsewhere.
Evaluating Determinants of 2 x 2 Matrices
Identity Matrix
Definition 3: If A and X are n x n matrices, and AX = XA = I, then X is the multiplicative inverse of A, written A-1.
Evaluating Determinants of 2 x 2 Matrices
Show that the matrices are multiplicative inverses.
Example 1
Show that the matrices are multiplicative inverses.
Example 2
Definition 4: The determinant of a 2 x 2 matrix is ad – bc.
Determinant of a 2 x 2 Matrix
dc
ba
detA = = ad - bc
Symbols for the Determinant
Evaluate each determinant.
Example 3
Evaluate each determinant.
Example 4
Evaluate each determinant.
Example 5
Evaluate the determinant.
Does this matrix have an inverse?
TOTD
Let . If det A = 0, then A has no inverse.
If det A ≠ 0, then
Property: Inverse of a 2 x 2 Matrix
Aa b
c d
A 1 1
det A
d b c a
1
ad bcd b c a
Example 6 Determine whether each matrix has an
inverse. If an inverse matrix exists, find it.
Example 7 Determine whether each matrix has an
inverse. If an inverse matrix exists, find it.
Determine whether each matrix has an inverse. If an inverse matrix exists, find it.
TOTD
Quiz 4.1-4.3 Review
Determinant = detA = = ad – bc
If detA 0, then:
OR… in calculator: [A]-1
4.5 Review
Using Inverse Matrices to Solve Equations
AX = B
A-1(AX) = A-1B
(A-1A)X = A-1B
IX = A-1B
X = A-1B
Solve each matrix equation in the form AX = B.
Example 8
Solve each matrix equation in the form AX = B.
Example 9
Communications The diagram shows the trends in cell phone ownership over four consecutive years.
Write a matrix to represent the changes in cell phone use.
In a stable population of 16,000 people, 9927 own cell phones, while 6073 do not. Assume the trends continue. Predict the number of people who will own cell phones next year.
Use the inverse of the matrix from part (a) to find the number of people who owned cell phones last year.
Example 10
Solve the matrix equation in the form of AX=B.
TOTD
Recommended