Transcript 3.8 – Use Inverse Matrices to Solve Linear Systems

3.8 – Use Inverse Matrices to Solve
Linear Systems
The n x n identity matrix is a matrix with 1’s on
the main diagonal and 0’s elsewhere.
If A is any n x n matrix and I is the n x n
identity matrix, then AI = A and IA = A.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Two n x n matrices A and B are inverses of each
other if their product (in both orders) is the n x n
identity matrix. That is AB = I and BA = I. An n x n
matrix A has an inverse if and only if det A does not =
0. The symbol for the inverse of A is A-1.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 1:
Find the inverse of the matrix.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 2:
Solve the matrix equation AX = B for the 2 x 2
matrix X.
3.8 – Use Inverse Matrices to Solve
Linear Systems
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 3:
Use an inverse matrix to solve the linear
system.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 4:
Use an inverse matrix to solve the linear
system.
-2x + 3y = -11
5x + y = 19
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 5:
Use an inverse matrix to solve the linear
system.
4x - y = 10
-7x – 2y = -25
3.8 – Use Inverse Matrices to Solve
Linear Systems
Inverse of a 3 x 3 matrix
The inverse of a 3 x3 matrix is difficult to
compute by hand…use your calculator to
compute this inverse.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 6:
Use a graphing calculator to find the inverse
of A. Then use the calculator to verify your
results.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 7:
Use a graphing calculator to find the inverse
of A. Then use the calculator to verify your
results.
3.8 – Use Inverse Matrices to Solve
Linear Systems
Example 8:
A company sells three types of movie gift
baskets. A basic basket with 2 movie passes
and 1 package of microwave popcorn costs
$15.50. A medium basket with 2 movie
passes, 2 packages of popcorn, and 1 DVD
costs $37. A super basket with 4 movie
passes, 3 packages of popcorn, and 2 DVD’s
costs $72.50. Find the cost of each in the gift
baskets.