1. Chapter 7 Matrix Mathematics
Matrix Operations
Chapter 7 Matrix Mathematics
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2. Matrix Mathematics
Matrices are very useful in engineering calculations. For example, matrices are used to:
Efficiently store a large number of values (as we have done with arrays in MATLAB)
Solve systems of linear simultaneous equations
Transform quantities from one coordinate system to another
Several mathematical operations involving matrices are important
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3. Review: Properties of Matrices
A matrix is a one-or two dimensional array
A quantity is usually designated as a matrix by bold face type: A
The elements of a matrix are shown in square brackets:
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4. Review: Properties of Matrices
The dimension (size) of a matrix is defined by the number of rows and number of columns
Examples:
3 × 3: ×4:
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5. Review: Properties of Matrices
An element of a matrix is usually written in lower case, with its row number and column number as subscripts:
In MATLAB, an element is designated by the matrix name with the row and column numbers in parentheses: A(1,2)
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6. Matrix Operations Matrix Addition
Multiplication of a Matrix by a Scalar
Matrix Multiplication
Matrix Transposition
Finding the Determinant of a Matrix
Matrix Inversion
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7. Matrix Addition Vectors must be the same size in order to add
To add two vectors, add the individual elements:
Matrix addition is commutative:
A + B = B + A
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8. Multiplication of a Matrix by a Scalar
To multiple a matrix by a scalar, multiply each element by the scalar:
We often use this fact to simplify the display of matrices with very large (or very small) values:
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9. Multiplication of Matrices
To multiple two matrices together, the matrices must have compatible sizes:
This multiplication is possible only if the number of columns in A is the same as the number of rows in B
The resultant matrix C will have the same number of rows as A and the same number of columns
as B
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10. Multiplication of Matrices
Consider these matrices:
Can we find this product?
What will be the size of C?
Yes, 3 columns of A = 3 rows of B
2 X 2: 2 rows in A, 2 columns in B
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11. Multiplication of Matrices
Easy way to remember rules for multiplication:
These values must match
Size of Product Matrix
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12. Multiplication of Matrices
Element ij of the product matrix is computed by multiplying each element of row i of the first matrix by the corresponding element of column j of the second matrix, and summing the results
This is best illustrated by example
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13. Example – Matrix Multiplication
Find
We know that matrix C will be 2 × 2
Element c11 is found by multiplying terms of row 1 of A and column 1 of B:
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14. Example – Matrix Multiplication
Element c12 is found by multiplying terms of row 1 of A and column 2 of B:
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15. Example – Matrix Multiplication
Element c21 is found by multiplying terms of row 2 of A and column 1 of B:
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16. Example – Matrix Multiplication
Element c22 is found by multiplying terms of row 2 of A and column 2 of B:
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17. Example – Matrix Multiplication
Solution:
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18. Practice Problems Find C = AB
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19. Practice Problems Find C = AB
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20. Practice Problems Find C = AB
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21. Matrix Multiplication
In general, matrix multiplication is not commutative:
AB ≠ BA
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22. Transpose of a Matrix
The transpose of a matrix by switching its row and columns
The transpose of a matrix is designated by a superscript T:
The transpose can also be designated with a prime symbol (A’). This is the nomenclature used in MATLAB
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23. Determinant of a Matrix
The determinant of a square matrix is a scalar quantity that has some uses in matrix algebra. Finding the determinant of 2 × 2 and 3 × 3 matrices can be done relatively easily:
The determinant is designated as |A| or det(A)
2 × 2:
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24. Determinant of a Matrix
Examples:
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25. Determinant of a Matrix
3 × 3:
Similar for larger matrices, but easier to do with MATLAB or Excel
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26. Inverse of a Matrix Some square matrices have an inverse
If the inverse of a matrix exists (designated by -1 superscript), then
where I is the identity matrix – a square matrix with 1’s as the diagonal elements and 0’s as the other elements
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27. Inverse of a Matrix The inverse of a 2X2 matrix is easy to find:
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28. Inverse of a Matrix Example: find inverse of A:
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29. Check Result
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30. Practice Problem Find A-1, check that A A-1 = I
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31. Inverse of a Matrix
Note from the formula for the inverse of a 2 × 2 matrix that if the determinant equals zero, then the inverse is undefined
This is true generally: the inverse of a square matrix exists only of the determinant is non-zero
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