Difference between revisions of "Matrix multiplication MA265F12Alvarado" - Rhea

Revision as of 11:33, 13 December 2012

Matrix Multiplication and coordinate systems:

$ A=\left[\begin{array}{cccc}1&2&3&4\\5&6&7&8\end{array}\right] $ $ B=\left[\begin{array}{cccc}1&2\\5&6\\3&4\\7&8\end{array}\right] $ $ C=\left[\begin{array}{cccc}1&2\\3&4\end{array}\right] $

Within this section and given the matrices A, B, and C, you will be shown how to multiply matricies.

Step 1: Determining if Two Matrices Can Be Multiplied

To determine if two matrices can be multiplied, you must first look the dimension of each matrix. For instance, if we wanted to perform the operation AB, we need to look at the dimensions of A and B.

A is a 2x4 matrix and B is a 4x2 matrix. To see if you can multiply these matrices, place their dimensions next to each other in the order of the operation: AB = (2x4)(4x2). Now look at the inside dimension. If the inside dimension is the same, then you can multiply the matrices. So, in our case, we can perform the operation AB.

Question: Can you perform the operations BA, BC, CB, AC, or CA?

Answers:

BA: (4x2)(2x4), YES BC: (4x2)(2x2), YES CB: (2x2)(4x2), NO AC: (2x4)(2x2), NO CA: (2x2)(2x4), YES

Step 2: Determining Size of the Resultant

To determine the size of the resultant matrix multiplication, look at the dimensions of the matrices in the order of the operation. As an example, we will continue to look at the operation AB: (2x4)(4x2). Now look at the outside dimensions for the dimensions of the resultant matrix. Our new matrix will have dimensions (2x2)

Question: What are the dimensions of BA, BC, and CA?

Answers: BA: (4x2)(2x4), (4x4) BC: (4x2)(2x2), (4x2) CA: (2x2)(2x4), (2x4)

Step 3: Multiplying the Matrices

To multiply compatible matrices such as AB, multiply B's individual rows by all of A's columns. Every row of B multiplied by all of A's columns will give us our (2x2) resultant matrix