Download inverse matrix

Lesson 7.6 & 7.7
Inverses of a Square Matrix
&
Determinant
In matrices, the word inverse mean something familiar:
MATRIX • INVERSE MATRIX = IDENTITY MATRIX
1
A  A  In
2 x 2 Identity Matrix:
1 0 
I2  

0
1


You can think of the inverse matrix as the inverse operation
much like multiplication and division with real numbers
This gives us another way to solve systems:
2 x  3 y  z  1

3x  3 y  z  1
2 x  4 y  z  2

Set this system up with a variable matrix:
2 3 1  x    1
3 3 1  y    1 

   
2 4 1  z   2
Let
A = the variable coefficient matrix
X = the variable matrix
B = the constant matrix
Then we could simply write
AX = B
Then solve for X by multiplying both sides by the inverse of A
A  A1 X  A1B
Matrix A and its inverse cancel, and the result of the right side is
our answer.
X  A1B
Solving on the calculator:
Just use the x-1 key with matrix A
Example – solve the system
 x  y  z  1

3x  5 y  4 z  2
3x  6 y  5 z  2

Note:
We could find the inverse by hand by using an augmented matrix consisting of
the coefficient matrix and the identity matrix. Then you would use row
operations to change the coefficient matrix (left side) into the identity matrix.
The new matrix (right side) will be the inverse.
1  1 0 
A  1 0  1
6  2  3
1  1 0
 1 0  1
6  2  3
1 0 0
0 1 0
0 0 1
The Determinant
A useful number found in patterns when solving systems
Consider it an operation in matrices – it is built in the
calculator
Given matrix A
 a1
A
 a2
b1 

b2 
Then the determinant is:
det A  a1b2  a2b1
“Top diagonal product – bottom
diagonal product”
Easy to calculate for
a 2 x 2 actually. If
you are just asked
to find the
determinant I
wouldn’t use the
calculator.
Symbol for determinant is 2 vertical bars, like absolute value.
Example: Find A
0 2 1 
A  3  1 2
4 0 1
I would use a calculator for this one!