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10.4 Matrix Algebra. 1. Matrix Notation. A matrix is an array of numbers. Definition : The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns. 2. Sum and Difference of 2 matrices. To add/subtract… add corresponding elements. Evaluate:. - PowerPoint PPT Presentation
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10.4 Matrix Algebra
1. Matrix NotationA matrix is an array of numbers.
Definition: The Dimension of a matrix is m x n “m by n” where m = # rows, n = #columns
41
32
03
A
2. Sum and Difference of 2 matrices
To add/subtract… add corresponding elements.
Evaluate:
1390
A
0182
B
BA
Note: The matrices must be same dimensions!
3. Scalar Multiplication
We can multiply matrix by a number (known as scalar).
Example:
Find
1390
A
A2
4. Matrix MultiplicationMultiplication : Row-by-Column multiplication
Determine
dcba
A
hgfe
B
AB
Evaluate
4. Matrix Multiplication
AB
1390
A
0182
B
AB
4. Matrix Multiplicationmore practice:
6572
A
73101
B
AB 1)
ABA 3 2)
4. Matrix Multiplication
Dimensions: rows columns rows columns
AB will have dimensions
nmA
Important: For Matrix multiplication to work:
The number of columns in first matrixmust equal
number of rows in second!
pnB
AB Determine
4321
A
51
31
20
B
pm
Why is the product BA not possible?
4. Matrix Multiplication
Evaluate the following:
31
10
21C
51
31
20
B
BC 1)
CB 2)
100010001
D
BD 3)
5. Identity MatrixReal Numbers: 1 is the multiplicative identity. Example
Matrices: is the Multiplicative identity of a matrix ,
a square matrix with 1’s on diagonal, 0’s elsewhere.
is used to represent the order n (dimension)
Example: Order 2 Order 3
A matrix times its identity returns the original matrix.
1001
2I
100010001
3I
nI
AAI
aa 1
I
6. Inverse of a MatrixReal Numbers: Multiplicative Inverse of is (for any )
Matrices: Multiplicative Inverse of a matrix is a matrix read as: “A-inverse”
with the property:
1A
IAA
IAA
1
1
Definition:If a matrix does not have an inverse, it is called singular
a a/10a
11
aa
A
6. Inverse of a Matrix
Example:
Given and its inverse
show and
1213
A
IAA 1
32111A
IAA 1
IAA
IAA
1
1
6. b) Finding the Inverse of a MatrixTo 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 |
3512
A
1A
6. b) Finding the Inverse of a Matrix
Example:Find the multiplicative inverse of
Graphing calculator: To Enter Matrix data:
• 2nd MATRIX: Edit (Enter)• Dimensions 3 x 3
To find Inverse:• 2nd MATRIX: NAMES 1:[A] Enter “^-1”.
310054111
A
7. Solve a system of linear equations Inverse Matrix method
A system can be written using matrix notation:
A is the coefficient matrixB is the constant matrixX represents the unknowns.
Example: Write this system using matrix notation:
BAX
231541
zyyxzyx
7. Inverse matrix method
If has a unique solution
then is the solution.
Solve:
BAX
BAX 1
231541
zyyxzyx
7. Solve a linear system using inverse Matrix
Example:Solve the system:
Note: We found in an earlier example
231541
zyyxzyx
1A
7. Solve a linear system using inverse Matrix
Your turn:Solve the system:
6532
62
yxzyx
zx