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. So row row 1, column 1 of the the new matrix will be: 1(1)+2(5)+1(2)+2(6)+1(3)+2(7)+1(4)+2(8) = 62 (multiply 1 the 1st row of A and 2 by the 2nd row of A)

Next use B's second row (5 and 6) and multiply 5 across A's 1st row and 6 across A's 2nd row adding the resultants to get the 1st row and 2nd column value of the new matrix. (The resultant value you should get is 206.)


Question: What are the values of the 2nd row, 1st column and 2nd row, 2nd column?


Answers:

2nd row, 1st column: 134

2nd row, 2nd column: 278


Step 4: The Resultant

The resulting matrix of AB is :

$ C=\left[\begin{array}{cccc}62&206\\134&278\end{array}\right] $

Congratulations! :)

Properties of Matrix Multiplication

Matrices within the properties are not the same as those presented above

1. AB does NOT equal BA

2. A(BC) = (AB)C

3. A(B+C) = AB + AC

4. (A+B)C = AC + BC

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett