Have we ever seen this phenomenon before?
130
21312
185
100
010
001
130
21312
185
Let’s do some quick multiplication…
130
21312
185
100
010
001
130
21312
185
What do we call this?
5 • __= 5the
(multiplicative) identity
1
Let’s do some quick multiplication…
130
21312
185
100
010
001
130
21312
185
What will we call this?
Identity Matrix
I
Let’s do some quick multiplication…
Identity MatrixIdentity Matrix
AI= AIA= A
Any matrix multiplied by the identity matrix will be equal to itself.
Inverse MatrixInverse Matrix
A * A-1= I
Any matrix multiplied by its inverse will be the identity matrix.
Matrix EquationMatrix Equation
6
8
21
45
y
x
A linear system can be written as a matrix equation AX=B
Coefficient matrix Variable
matrix
Answer matrix
5 4 8
1 2 6
x y
x y
Suppose ax = b
How do you solve for x?
We cannot divide matrices, but we can multiply by the inverse.
AX = BAA-1-1 AA-1-1
IX = AA-1-1B
X = AA-1-1B
Solving Matrix EquationsSolving Matrix Equations
Ex. 2Ex. 2 Solve using matrices. Solve using matrices.
3 4 5
2 10
x y
x y
x = -7
y = -4
10
5
12
43
y
x
A B
X = AA-1-1B
AX = B
(-7, -4)
Ex. 4Ex. 4 Solve using matrices Solve using matrices
2 3 1
3 3 1
2 4 2
x y z
x y z
x y z
x = 2
y = -1
z = -2
(2, -1, -2)
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