1. Fundamentals of Engineering Analysis
EGR Matrix Multiplication, Types

2. Matrix Multiplication
Define “Conformable”
To multiply A * B, the matrices must be conformable.
Given matrices: A m x n and B n x p
The number of “Columns” n of A, must equal the number of “Rows” n of B
Which defines the order of the multiplication A=
2 x 3
For A: m=2, n=3 B=
3 x 4
For B: n=3, p=4
For A * B, n=n; i.e. 3=3, so A*B is “conformable”
Note that B * A is Undefined (not allowed) because p = m

3. Order of Multiplication
The order in which a multiplication is expressed is important.
We use the terms “pre-multiply” or “post-multiply” to stipulate the order.
Given A * B = C, we say that “B” is “pre-multiplied” by “A”
(we could also say that A is post-multiplied by B).
Because matrices must be conformable for multiplication; in general
A * B = B * A
In other words, Matrix Multiplication is NOT Commutative
(except in special cases)

4. Matrix Multiplication
Is a Row on Column operation
n x p
3 x 3
m x n
2 x 3 =
A * B = C is Conformable
m x p
The Product C will be a 2 x 3

5. * = Matrix Multiplication
C11 is made up of Row 1 from A, and Column 1 from B
Note the “sum of products” form
C12 is made up of Row 1 from A, and Column 2 from B
Remember:

6. Matrix Multiplication
A * B =
2 x 2
B * A =
3 x 3

7. Matrix Multiplication
A * B = 9 5 * = 1 7

8. = * B * A = 4 3 -1 2 1 1 4 -1 11 Matrix Multiplication
Work this out yourself, before proceeding,
To make sure you understand the method of matrix multiplication.

9. * ax1 + bx2 + cx3 = d Linear Systems as Sum of Products
Sum of Products form
ax1 + bx2 + cx3 = d x1 x2 x3
[ a b c ] - a 1 x 3 row vector
- a 3 x 1 column vector
[ a b c ] * x1 x2 x3
= [ d ]
- a 1 x 1 scalar – i.e.;
ax1 + bx2 + cx3 = d

10. A * B * C = not conformable
Conformability and Order of Matrix Multiplication
A5x4
B4x5
C6x4
Given:
A * B = D5x5
B * A = E4x4
A * C = not conformable
C * A = not conformable
C * B = F6x5
A * B * C = not conformable
C * B * A = G6x4

11. * = * = Properties of a Zero Matrix
In Algebra, x * 0 = 0, but if x = 0, and y = 0, then x * y = 0
In Matrix Algebra, even if A = 0, and B = 0, A * B can be [0] * =
Note that:

12. ? = * Matrix Form of Linear Equations
Distributive Property: A(B+C) = AB + AC
Associative Property: A(BC) = (AB)C
Then
can become ?
How do we
solve this
system
of equations = *
Any Order A

13. Rule: The Row becomes the Column, and the Column becomes the Row
Special Matrices
The Transpose Matrix
Rule: The Row becomes the Column, and
the Column becomes the Row
A is a 2x3, so AT will be a 3x2
For a 3x3

14. Properties of the Transpose Matrix
A*B=
AT*BT = ?
BT*AT =
(AB)T= BT*AT

15. (A+B)T = AT + BT (AB)T = BTAT Additional Properties of the Transpose
If A+B and A*B are allowed (are conformable), then
(A+B)T =
AT + BT
(AB)T = BTAT

16. A + AT must also be Symmetric
The Symmetric Matrix
The Diagonal
Must be Square: n x n
A = AT
A + AT must also be Symmetric

17. All off-diagonal elements
The Diagonal Matrix
All off-diagonal elements
Are Zero
Must be Square: n x n
If A and B are
Diagonal +
A+B
will be Diagonal =
If A and B are
Diagonal *
A*B
will be Diagonal =

18. Amxn * In = A or Im * Amxn = A
The Identity Matrix
Must be Square: n x n
And must be Diagonal
Can be any Order
Notation: IN
The Unity term
I*A = A
A*I = A
A does not have to be square
Amxn * In = A or Im * Amxn = A

19. A * A = A2 for Square Matrices Only
Powers of Matrices
A * A = A2 for Square Matrices Only
A * A2 = A3 … and so on
If A is Diagonal …
A2 = a112, a222, a332 = *

20. Matrix Math on the TI-89 Calculator
My Philosophy for using Calculators
(and Computers …)
Be aware of the
Order of Magnitude
Sign Errors are easy to miss
Double check your work
If you understand the solution methodology,
You will understand the answer.

21. Matrix Math on the TI-89 Calculator
A*B – not conformable
B*A = ?

22. Matrix Math on the TI-89 Calculator (cont.)

23. Matrix Math on the TI-89 Calculator (cont.)

24. Using the Matrix Editor on the TI-89

25. Questions?