Question

Explain why the number of columns in matrix A must equal the number of rows in matrix B when finding the product AB.

Short answer

Finding the product of two matrices is only possible when the inner dimensions are the same, meaning that the number of columns of the first matrix is equal to the number of rows of the second matrix.

Step-by-step solution

Step 1. Consider an example

Let us consider two examples to explain why the number of columns in A should be equal to the number of rows in B. In the first example, let us take the number of columns in A equal to the number of rows in B, and in the second example when the number of columns in A is not equal to the number of rows in B.

Step 2. Explanation using example

Here the number of columns in A is equal to the number of rows in B. The first entry of matrix A B is which is the multiplication of Row 1 of A with Column 1 of B.

Similarly, we can perform the rest of the multiplication.

Now we can see that number of columns in A is not equal to the number of rows in B. The symbol of the question mark shows that we have no quantity in matrix B to whom we can multiply. Therefore the number of columns in A must be equal to the number of rows in matrix B for multiplication of matrices.