Fundamentals of Engineering Analysis EGR 1302 - Matrix Multiplication, Types

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

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)

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

* = 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:

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

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

= * 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.

* 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

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

* = * = 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:

? = * 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

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

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

(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

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

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 =

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

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 = *

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.

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

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

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

Using the Matrix Editor on the TI-89

Questions?