How to Add Matrix Multiplication?

Matrix multiplication isn't quite "adding" in the typical sense, but here's how matrix multiplication works, breaking down the process step-by-step based on the provided reference:

Understanding Matrix Multiplication

Matrix multiplication involves combining elements from two matrices to produce a new matrix. The key is understanding how rows and columns interact.

Steps for Matrix Multiplication

Here's how to multiply two matrices, utilizing the row-column approach:

1. Check Dimensions: Ensure that the number of columns in the first matrix equals the number of rows in the second matrix. If this condition isn't met, matrix multiplication is not possible. If matrix A is of dimensions m x n, and matrix B is of dimensions p x q, then n must equal p for the multiplication to be defined. The resulting matrix will have dimensions m x q.

2. Row and Column Focus: Focus on the first row of the first matrix and the first column of the second matrix.

3. Pairwise Multiplication: Multiply the first number in the row by the first number in the column, the second number in the row by the second number in the column, and so on (as mentioned in the YouTube video "How to organize, add and multiply matrices - Bill Shillito").

4. Sum the Products: Add up all the products calculated in the previous step. This sum becomes the element in the first row and first column of the resulting matrix.

5. Repeat: Continue this process, moving across the rows of the first matrix and down the columns of the second matrix to fill in each element of the resulting matrix.

Example

Let's say you have two matrices:

Matrix A:

| 1  2 |
| 3  4 |

Matrix B:

| 5  6 |
| 7  8 |

To find the element in the first row and first column of the result:

(1 5) + (2 7) = 5 + 14 = 19

To find the element in the first row and second column of the result:

(1 6) + (2 8) = 6 + 16 = 22

And so on...

Resulting Matrix:

| 19  22 |
| 43  50 |

Key Considerations

• The order of multiplication matters: A * B is generally not equal to B * A.
• Matrix multiplication is associative: (A * B) * C = A * (B * C)