Download The inverse of a matrix

Name
Date _______________________________
Honors Algebra 2
Lessons 4.3-4.4 notes and homework
The inverse of a matrix
Comparing number multiplication and matrix multiplication
Identity element
Property of the
identity element
What it means
for 2 elements to
be inverses
Inverse of an
element
number multiplication
1 is the identity element.
matrix multiplication (using n-by-n square matrices)
For any x, 1·x = x and x·1 = x.
For any n-by-n matrix A, IA = A and AI = A.
Two numbers x and y are called
inverses if xy = 1.
Two matrices A and B are called inverses if
AB = I and BA = I.
The inverse of x is
This means x ·
1
x
1
x
I=
.
= 1 and
1
x
· x = 1.
Does everything
have an inverse?
0 is the only number without an
inverse.
How to solve a
typical equation
using inverses
To solve: ax = b.
Multiply both sides by
1
a
ax =
1
a
b
1x =
1
a
b
x=
1
a
:
1
0

0


0
0 0  0
1 0  0

0 1  0

   
0 0  1
is the identity element.
The inverse of A is named A–1.
This means A · A–1 = I and A–1 · A = I..
There are some square matrices A for which an A–1
does not exist. Such matrices are called singular or
non-invertible.
To solve: AX = B where A, X, B are matrices.
Multiply both sides by A–1 on the left:
A–1AX = A–1B
I X = A–1B
X = A–1B
b
a
Ways to find the inverse of a matrix
There are several possible ways to get the inverse A–1 of a matrix A:
 On your calculator: Enter matrix [A], then calculate [A]–1 (you must use the  key for the -1).
[See page 236 Example 2, and the calculator instructions on page 269.]
 Using row operations: Start with matrix A alongside the identity matrix, like this: [A | I].
Do the usual row operation procedure to get the reduced form. Now you should see the identify
matrix on the left, and whatever is on the right will be A–1. [A | I]  [I | A–1].
row ops.


By setting up and solving systems of equations: See page 236 (top) for an example.
By working out a specific formula for the 2-by-2 case: See page 240 challenge problem 46.
New way to solve a system of equations using an inverse matrix
Write the system as a matrix equation AX = B where A is a matrix of coefficients, X is a matrix of
variables, and B is a matrix of constants. Then use X = A–1B to get the solution.
Example: [See page 246 for a fuller explanation; page 270 for calculator instructions.]
Problem
5x + 2y – z = –7
x – 2y + 2z = 0
3y + z = 17
Write in AX = B form…
5 2  1  x    7
1  2 2    y    0 

    
1   z   17 
0 3
Rewrite as X = A–1B…
1
 x  5 2  1   7
 y   1  2 2    0 
  
  
1   17 
 z  0 3
Answer
 x    2
 y   4 
   
 z   5 
Name
Date _______________________________
Honors Algebra 2
Lessons 4.3-4.4 notes and homework
Homework
Some parts of this assignment refer to problems from textbook lessons 4.3–4.4.
A. Do these problems where you determine whether two matrices are inverses of each other:
4.3 # 10–12. Hint: Multiply the matrices; do you get 1 0 as the result?
0 1 


B. Find inverse matrices using your calculator: 4.3 # 21, 23, 37.
Calculator help: p. 236 Example 2; calculator instructions on p. 269.
C. Find the inverse from #21 again using a row operation procedure. Here’s how:

Set up the matrix alongside the identity matrix, like this:
  2 2  1 1 0 0
 3  5 4 0 1 0


 5  6 4 0 0 1

Use row operations to put the above matrix into reduced form.
Your choice: Do row operations by hand or use the Matrix Row Operations Tool linked
from our homework web page.
The result should look like:
1 0 0 ? ? ?
0 1 0 ? ? ?


0 0 1 ? ? ?

Whatever numbers appear on the right side give you the inverse matrix.
D. Repeat # 23 and 37 using the row operation procedure. (Something different happens in #23;
explain it.)
E. Look at the method for finding the inverse of
1 2
3 5


shown at the top of page 236.
It involves setting up and solving systems of equations.
 1 0
Now, find the inverse of  2 1  using the same method.
1 4 
F. Do 4.3 # 46.
G. Do 4.3 # 69.
H. Read New way to solve a system of equations using an inverse matrix on the front of this
sheet. If you need more explanation, see textbook section 4.4, especially p. 246 Example 2.
I. Do 4.4 # 11, 15, 17.
J. Do 4.4 # 19–25 odd, using the AX = B  X = A–1B method.
K. Optional challenge: 4.4 # 30.