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

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.

Examples 1 & 2

Example 1

𝐴=[2 31 2]𝐵=[ 2 −3

−1 2 ]

Example 2

𝐴=[−2 −5−3 −8]𝐵=[−8 5

3 −2]

Definition 4: The determinant of a 2 x 2 matrix is ad – bc.

Determinant of a 2 x 2 Matrix

a b

c d

Evaluate each determinant.

Examples 3 - 5

Example 3

𝑑𝑒𝑡 [−3 42 −5 ]

Example 4

𝑑𝑒𝑡 [2 −33 −2 ]

Example 5

𝑑𝑒𝑡 [𝑎 00 𝑎 ]

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

Examples 6 & 7 Determine whether each matrix has an

inverse. If an inverse matrix exists, find it.

Example 6

𝑀=[−2 25 −4]

Example 7

𝑁=[3 92 6]

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.

Examples 8 & 9

Example 8

[−2 −51 3 ] 𝑋=[−22 ]

Example 9

[3 −44 −5 ] 𝑋=[0 −22

0 −28 ]

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