Transcript Document
Inversion
9.7 The Inverse of a Matrix
Objective: Students will solve systems using matrix
algebra, inverse matrices and identity matrices.
Matrix Inversion
1
1
B B BB
Like a reciprocal
in scalar math
I
Like the number one
in scalar math
Using matrix equations
Identity matrix: Square matrix with 1’s on the diagonal
and zeros everywhere else
1 0
0 1
1
0
0
0
1
0
2 x 2 identity matrix
0
0
1
3 x 3 identity matrix
The identity matrix is to matrix multiplication as
1 is to regular multiplication!!!!
___
Multiply:
1 0 5 2
3 4 =
0 1
5 2
3
4
1 0
=
0 1
5 2
3
4
5 2
3
4
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically,
IA = A and AI = A !!
Using matrix equations
Inverse Matrix: 2 x 2
A
a b
c d
1
A
1 d b
ad bc c a
In words:
•Take the original matrix.
•Switch a and d.
•Change the signs of b and c.
•Multiply the new matrix by 1 over the determinant of the original matrix.
Using matrix equations
Example: Find the inverse of A.
A
1
A
1
A
4
2
4 10
10 4
1
2
(2)(10) ( 4)(4) 4
1 10 4
2 =
4 4
5
1
2
1
1
2
Find the inverse matrix.
8 3
5
2
Inverse =
Matrix A
1 Matrix
det Reloaded
Det A = 8(2) – (-5)(-3) = 16 – 15 = 1
=
1
1
2 3
5 8
=
2 3
5 8
What happens when you multiply a matrix by its inverse?
1st:
What happens when you multiply a number by its inverse?
A & B are inverses. Multiply them.
8 3 2 3
5
2 5 8
=
So, AA-1 = I
1 0
0 1
7
1
7
Why do we need to know all this? To Solve Problems!
Solve for Matrix X.
8 3
5
2
X
4 1
3
1
=
We need to “undo” the coefficient matrix.
2 3 8 3
5 8 5
2
1 0
0 1
X
X
X
=
=
Multiply it by its INVERSE!
2 3 4 1
5 8 3
1
1 1
4 3
1
=
1
4 3
Using matrix equations
You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8
becomes
3 2
2 1
x
11
y
= 8
Coefficient Variable
matrix
matrix
Answer matrix
Using matrix equations
Example: Solve
3 2
2 1
x
y
11
= 8
for x and y .
Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.
1
A
3 2 1
2 1
1 1 2
=
1 2 3
2
1
2 3
2
1
=
2 3
2
3 2 x 1
y =
2 1 2 3
1 0 x 5
y = 2
0 1
x 5
y = 2
11
8
Using matrix equations
Wow!!!!
x = 5; y = -2
It works!!!!
Check:
3x + 2y = 11
3(5) + 2(-2) = 11
2x + y = 8
2(5) + (-2) = 8
You Try…
Solve:
4x + 6y = 14
2x – 5y = -9
(1/2, 2)