1. WHAT IS MATRIX MULTIPLICATION?
Matrix multiplication is a theorem that allows for us to multiply two matrices of different dimensions together; there are rules to which matrices can be multiplied together.
- Matrix multiplication theorem
- This theorem explains how one is to go about multiplying matrices and their entries, this will be expanded upon after explaining the rules of matrix multiplication and why it is such a nifty method.
2. DIMENSIONS WHILE MULTIPLYING
- Imagine we have a matrix denoted by A and A is an m x n matrix (m x n simply means that the dimensions of matrix a do not have the same number of rows as columns. I.e. 3 x 2 or 17 x 5)
- Imagine we have another matrix denoted by B which is an n x p matrix.
- Notice how the dimensions of the two matrices are off and are not a square, the multiplication of these two matrices does not inherently seem possible. However, these matrices can be multiplied because the columns of A (m x n) are equivalent to the rows of matrix B (n x p) . This equivalence allows all of the rows and columns to be accounted for whilst multiplying.
- The resulting product of these two matrices ends up having a dimension comprised of the rows from A (m x n) and the columns from B (n x p).
-Therefore AB produces an m x p matrix
3. RULES TO CONSIDER WHILE MULTIPLYING
The dimensions of the two matrices do not need to be equal to multiply. Rather, the columns of the first matrix must be equal to the amount of rows in the second matrix.
Matrix multiplication does NOT follow the commutative property.
o Meaning AB is not equivalent to BA. If you are able to produce a matrix, the resulting product won't match the desired matrix. In most cases however, a matrix wont be able to be produced because the required columns and rows are not equivalent.
o In short, matrix multiplication is contingent upon dimensions.
4. MATRIX THEOREM.
Now that we understand the importance of dimensionality, we can now learn how to multiply matrices.
Review matrix theorem
This theorem is simply explaining how you take the product and summation of equivalent rows and columns.
I.e.
First be sure that the dimensions are correct. A is a 2x3 and B is a 3x2; they can be multiplied successfully.
Take the first row of A and multiply it by the first column.
o Do this until you have multiplied the first row by all of the column entries and in B.
Ensure the dimensions are correct
5. WHAT IS MATRIX MULTIPLICATION?
1. Associative
2. Not commutative
3. Distributive
4. A summation of rows multiplying columns
5. Not piecewise multiplication
… AND WHY DOES IT MATTER?
6. GRAPH THEORY
7. QUANTUM MECHANICS
8. MATHEMATICAL FINAANCE MODELS/COMPUTATIONS
PRACTICE PROBLEMS
CAN A…
A) 2X3 X (2X3) B) 2X1 X (1X3) C) 25X3 X (3X8) D) 4X3 X (3X2)
BE MULTIPLIED SUCCESSFULLY?
-GENERATE A MATRIX WITH ONE OF THE DIMENSIONS AND ATTEMPT TO MULTIPLY NORMALLY AMD THEN VIA THE COMMUTATIVE PROPERTY.
