Unlike adding or subtracting matrices, you don’t have to have the same dimensions when multiplying matrices. The one thing that you do have to have in common is that the first matrix has to have the same amount of columns as the second matrix has rows. So if the first matrix has 2 columns, than the second matrix must have 2 rows for you to be able to do multiplication.
When you multiply matrices together, one thing that is always guaranteed is that the resulting matrix will have the same amount of rows as the first matrix, and the same amount of columns as the second matrix. Another thing that is always guaranteed is that if you multiple matrix A by Matrix B, it’s not going to be the same thing as multiplying matrix B by matrix A
So for example, let’s multiply [ 1 2] by [ 3 ].
[ 4 ]
The first thing that we do is figure out the dimensions of our matrix. There’s one row in the first matrix and one column in the second matrix. That means that the matrix will have 1 row and 1 column. So this is what our matrix will look like: [ x ].
Our first step when we actually start multiplying is to multiply 1 by 3. We get 3. Then, we multiply 2 by 4. We get 8. Our final step is to add the two products, and our answer is a matrix like this: [ 11 ].
let’s do a harder one. Let’s do this:
[ 1 2] × [ 4 3] = [ 1×4 + 2×2 1×3 + 2×1 ] = [ 8 5 ]
[ 3 4] × [ 2 1] = [ 3×4 + 4×2 3×3 + 4×1 ] = [ 20 13 ]
So the rule for doing this is pretty simple. Multiply the first row of the first matrix by the first column of the second matrix. Then multiply the first row in the first matrix by the second row in the second matrix (if there is one). Do this until you’ve multiplied all of the columns in matrix 2 by the first row in matrix 1. Then do the same thing with the second row in matrix 1 (if there is one). This process continues until you’ve reached the final row in the first matrix.
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