Matrix Multiplication

Matrix multiplication falls into two general categories:

• Scalar in which a single number is multiplied with every entry of a matrix
• Multiplication of an entire matrix by another entire matrix For the rest of the page, matrix multiplication will refer to this second category.

In the scalar variety, every entry is multiplied by a number, called a scalar.

What is matrix multiplication?

Answer: You can multiply two matrices if, and only if, the number of columns in the first matrix equals the number of rows in the second matrix.

Otherwise, the product of two matrices is undefined.

In the picture on the left, the matrices can be multiplied since the number of columns in  the 1st one, matrix A, equals the number of rows in the 2nd, matrix B.

The Dimensions of the product matrix

Rows of 1^st matrix × Columns of 2^nd

4 × 3

If we multiply a 2×3 matrix with a 3×1 matrix, the product matrix is 2×1

Here is how we get M₁₁ and M₁₂ in the product.


Practice Problem


Is the product of matrix A and Matrix B below defined or undefined?

Answer

Since the number of columns in Matrix A does not equal the number of rows in Matrix B. The multiplication of A and B is undefined.