10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of...
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Transcript of 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of...
![Page 1: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/1.jpg)
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
![Page 2: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/2.jpg)
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
![Page 3: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/3.jpg)
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!
![Page 4: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/4.jpg)
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
![Page 5: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/5.jpg)
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
![Page 6: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/6.jpg)
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
![Page 7: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/7.jpg)
4. Matrix MultiplicationYour turn to practice:
65
72A
73
101B
AB 1)
ABA 3 2)
![Page 8: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/8.jpg)
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
![Page 9: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/9.jpg)
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
![Page 10: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/10.jpg)
5. Identity MatrixIdentity Property
Example:
Given the matrix:
AIA
AAI
23
41A
![Page 11: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/11.jpg)
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
![Page 12: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/12.jpg)
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
![Page 13: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/13.jpg)
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
![Page 14: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/14.jpg)
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
![Page 15: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/15.jpg)
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
![Page 16: 10.4 Matrix Algebra 1.Matrix Notation 2.Sum/Difference of 2 matrices 3.Scalar multiple 4.Product of 2 matrices 5.Identity Matrix 6.Inverse of a matrix.](https://reader035.fdocuments.in/reader035/viewer/2022071807/56649ecd5503460f94bda515/html5/thumbnails/16.jpg)
7. Solve a linear system using inverse Matrix
Your turn:Solve the system:
6
532
62
yx
zyx
zx