1. Matrix Mathematics in MATLAB and Excel
Chapter 7 Matrix Mathematics
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2. Review: 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 rows as B
Engineering Computation: An Introduction Using MATLAB and Excel

3. Review: Multiplication of Matrices
Easy way to remember rules for multiplication:
These values must match
Size of Product Matrix
Engineering Computation: An Introduction Using MATLAB and Excel

4. Review: 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
In general, matrix multiplication is not commutative:
AB ≠ B A
Engineering Computation: An Introduction Using MATLAB and Excel

5. Practice Problem Find C = A B (2 X 4) X (4 X 3) = (2 X 3)
Engineering Computation: An Introduction Using MATLAB and Excel

6. Review: Transpose
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
Engineering Computation: An Introduction Using MATLAB and Excel

7. Review: Determinate
The determinate of a square matrix is a scalar quantity that has some uses in matrix algebra. Finding the determinate of 2 × 2 and 3 × 3 matrices can be done relatively easily:
The determinate is designated as |A| or det(A)
2 × 2:
Engineering Computation: An Introduction Using MATLAB and Excel

8. Review: Inverse 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
The inverse of a square matrix exists only of the determinate is non-zero
Engineering Computation: An Introduction Using MATLAB and Excel

9. Matrix Operations in Excel
Excel has commands for:
Multiplication (mmult)
Transpose (transpose)
Determinate (mdeterm)
Inverse (minverse)
Important to remember that these commands apply to an array of cells instead of to a single cell
When entering the command, you must identify the entire array where the answer will be displayed
Engineering Computation: An Introduction Using MATLAB and Excel

10. Excel Matrix Multiplication
Repeat previous problem – first enter the two matrices:
Engineering Computation: An Introduction Using MATLAB and Excel

11. Excel Matrix Multiplication
Good practice to label the matrices:
Engineering Computation: An Introduction Using MATLAB and Excel

12. Excel Matrix Multiplication
Shading and borders help the matrices stand out:
Engineering Computation: An Introduction Using MATLAB and Excel

13. Excel Matrix Multiplication
Array of cells for the product must be selected – in this case, a 2 × 3 array:
Engineering Computation: An Introduction Using MATLAB and Excel

14. Excel Matrix Multiplication
The MMULT function has two arguments: the ranges of cells to be multiplied. Remember that the order of multiplication is important.
Engineering Computation: An Introduction Using MATLAB and Excel

15. Excel Matrix Multiplication
Using the Enter key with an array command only returns an answer in a single cell. Instead, use Ctrl + Shift + Enter keys with array functions
Engineering Computation: An Introduction Using MATLAB and Excel

16. Excel Matrix Multiplication
Answer cells formatted:
Engineering Computation: An Introduction Using MATLAB and Excel

17. Excel Transpose Use Ctrl + Shift + Enter to input command
Engineering Computation: An Introduction Using MATLAB and Excel

18. Excel Determinate
Since determinate is a scalar, select a single cell and use Enter to input command
Engineering Computation: An Introduction Using MATLAB and Excel

19. Excel Matrix Inversion
Remember that only square matrices can have inverses
Engineering Computation: An Introduction Using MATLAB and Excel

20. Excel Matrix Inversion
Ctrl + Shift + Enter to input command
Engineering Computation: An Introduction Using MATLAB and Excel

21. Check: A X A-1 = I, the identity matrix:
Engineering Computation: An Introduction Using MATLAB and Excel

22. Matrix Operations in MATLAB
Matrices are easily handled in MATLAB
We will look at commands for:
Multiplication
Transpose
Determinate
Inverse
Engineering Computation: An Introduction Using MATLAB and Excel

23. MATLAB Matrix Input
Recall that to enter matrices in MATLAB, the elements are enclosed in square brackets
Spaces or commas separate element within a row; semi-colons separate rows
>> G = [2 5 8; ; 3 1 2]
G =
Engineering Computation: An Introduction Using MATLAB and Excel

24. MATLAB Matrix Multiplication
>> G = [2 5 8; ; 3 1 2] G = >> H = [4 5; 6 12; 1 1] H =
Matrix multiplication symbol is same as for scalar multiplication (*):
>> L = G*H
L =
Engineering Computation: An Introduction Using MATLAB and Excel

25. Repeat Earlier Example
Find C = A B
>> A = [ ; ];
>> B = [2 3 1;0 1 0; ; 2 2 1];
>> C = A*B
C =
Engineering Computation: An Introduction Using MATLAB and Excel

26. MATLAB Transpose
The transpose command switches rows and columns of a matrix:
>> A = [10 6; 12 9; 1 3; 0 1]
A =
>> B = transpose(A)
B =
Engineering Computation: An Introduction Using MATLAB and Excel

27. MATLAB Transpose
Shortcut: The prime symbol (') after a matrix also performs the tranposition:
>> A = [10 6; 12 9; 1 3; 0 1]
A =
>> B = A'
B =
Engineering Computation: An Introduction Using MATLAB and Excel

28. MATLAB Determinate
The det command finds the determinate of a square matrix:
>> A = [2 5 6; ; 3 2 2]
A =
>> B = det(A)
B = 53
Engineering Computation: An Introduction Using MATLAB and Excel

29. MATLAB Inversion The inv command finds the inverse of a square matrix:
>> B = inv(A)
B =
Engineering Computation: An Introduction Using MATLAB and Excel

30. Check: >> C = A*B C = 1.0000 0 -0.0000 0.0000 1.0000 -0.0000
Product of a matrix and its inverse is the identity matrix
Engineering Computation: An Introduction Using MATLAB and Excel

31. MATLAB Matrix Inversion
Another example:
A =
>> B = inv(A)
Warning: Matrix is singular to working precision.
B =
Inf Inf Inf
Can you guess why this matrix has no inverse?
What is the determinate of A?
Engineering Computation: An Introduction Using MATLAB and Excel

32. MATLAB Matrix Inversion
Note that the third row of the matrix is simply the first row multiplied by a constant (2). When this happens, the matrix is singular and has no inverse. Also, notice that the determinate of the matrix is zero:
>> det(A)
ans =
Engineering Computation: An Introduction Using MATLAB and Excel

33. Linear Simultaneous Equations
Consider an equation with three unknowns x, y, and z
If the equation contains only linear terms of x, y, and z – that is, only constants multiplied by each variable – and constants, then the equation is linear
If the equation contains any terms such as x2, cos(x), ex, etc., then the equation is non-linear
Engineering Computation: An Introduction Using MATLAB and Excel

34. Linear Simultaneous Equations
Example of a linear equation:
With what we know about multiplying matrices, we can write this equation as:
(1 X 3) X (3 X 1) = (1 x 1)
Engineering Computation: An Introduction Using MATLAB and Excel

35. Linear Simultaneous Equations
Let’s add a second equation:
We can add this to our previous matrix equation by adding a second row to the first matrix and to the product:
(2 X 3) X (3 X 1) = (1 x 2)
Engineering Computation: An Introduction Using MATLAB and Excel

36. Linear Simultaneous Equations
Finally, let’s add a third equation:
Our matrix equation is now:
(3 X 3) X (3 X 1) = (1 x 3)
Engineering Computation: An Introduction Using MATLAB and Excel

37. Linear Simultaneous Equations
We will use matrix mathematics to solve the equations for x, y, and z
The first step in solving a series of linear simultaneous equations is to write them in matrix form
For n simultaneous equations and n unknowns:
where A is the coefficient matrix (n × n); X is the matrix of unknowns (n × 1), and C is the constant matrix (n × 1)
Engineering Computation: An Introduction Using MATLAB and Excel

38. Linear Simultaneous Equations
Recall that if there are more unknowns then equations, then we cannot find a unique solution
If there are more equations than unknowns, then some equations must be redundant
If there are exactly the same number of equations and unknowns, then there may be a unique solution. In this case the coefficient matrix will be square
Engineering Computation: An Introduction Using MATLAB and Excel

39. Practice Write these four equations in matrix form:
Engineering Computation: An Introduction Using MATLAB and Excel

40. Solution
In Chapter 8, we will learn how to find the unknowns a, b, c, and d with matrix methods
Engineering Computation: An Introduction Using MATLAB and Excel