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
