10.4 Matrix Algebra
1. Matrix Notation
2. Sum/Difference of 2 matrices
3. Scalar multiple
4. Product of 2 matrices
5. Identity Matrix6. Inverse of a matrix
a) Verify the inverse of a matrix b)Finding the inverse7. Solve a system using inverse matrices
1. Matrix Notation
Notation: refers to the element in row i, column j of a matrix A.
Notation: An “m x n” matrix has m rows and n columns
Example:
Identify the element
ija
23a
2720
854
321
4
5
0
A
2. Sum and Difference of 2 matrices
To add/subtract… add corresponding elements.
1)
2)
13
90A
01
82B
BA
BA
Note: The matrices must be same dimensions!
3. Scalar Multiplication
We can multiply matrix by a number (known as scalar).
kA implies the number k is multiplied times every element in A:
Example:
Find 1) 2)
13
90A
A2
01
82B
BA 23
4. Matrix MultiplicationMultiplication is NOT like addition (where we added corresponding elements).
You will NOT multiply corresponding elements.
Given:
Find the product:
13112A
3
4
0
1
B
Evaluate
4. Matrix Multiplication
AB
)0)(9()8)(0( AB
)1)(9()2)(0(AB
)1)(1()2)(3(
AB
)0)(1()8)(3(
AB
)0)(1()8)(3()1)(1()2)(3(
)0)(9()8)(0()1)(9()2)(0(AB
13
90A
01
82B
AB
4. Matrix MultiplicationYour turn to practice:
65
72A
73
101B
AB 1)
ABA 3 2)
4. Matrix Multiplication
rows columns rows columns
Example: is not possiblewhen columns in A does notequal rows in B:
nmA
Important: Matrix multiplication can only be performed if
The number of columns in first matrixis equal to
number of rows in second!
pnB
1
6
5
9
4
2
,53
11BA
AB
5. Identity MatrixDefinition: The identity Matrix is a square matrix thathas 1’s on diagonal and 0’s elsewhere
An identity matrix has the same properties as 1 in the real numbers.
10
012I
100
010
001
3I
5. Identity MatrixIdentity Property
Example:
Given the matrix:
AIA
AAI
23
41A
6. Inverse of a Matrix
The Inverse is the matrix A is and satisfies
Example:Given and its inverse
show and
12
13A
IAA 1
1A
32
111A
IAA 1
IAA
IAA
1
1
Definition:If a matrix does not have an inverse, it is called singular
6. b) Finding the Inverse of a Matrix
To find the inverse:
1) Form augmented matrix
2) Transform to reduced row echelon form (Gauss-Jordan).
3) The identity matrix will magically appear on the right hand side of the bar! This is
1A
Example:Find the multiplicative inverse of
Verify it when finished!
IA |
35
12A
1A
6. b) Finding the Inverse of a Matrix
Example:Find the multiplicative inverse of
Verify when finished!Your turn… Find the inverse for
310
054
111
A
310
054
111
A
7. Solve a system of linear equations using the inverse matrix method
If a system has a unique solution
where A is the coefficient matrix, X and B are 1 column matrices.
then is the solution.
1) Find 2) Multiply
3) The result in 2) is the solution
BAX
BAX 11A
BA 1
X
7. Solve a linear system using inverse Matrix
Example:Solve the system:
Note: We found in an earlier example
23
154
1
zy
yx
zyx
1A
7. Solve a linear system using inverse Matrix
Your turn:Solve the system:
6
532
62
yx
zyx
zx
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