Matrix Multiplication

How to multiply two matrices

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.

Scalar Matrix Multiplication

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

scalar matrix multipication picture

What is the answer to the scalar multiplication
problem below?

What is matrix multiplication?

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.

The product matrix's dimensions are $$ \rightarrow $$ (rows of first matrix) × (columns of the second matrix )

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

Example 1

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.

M11 = r11× t11  +  r12× t21  +   r13×t31
M12 = r21× t11  +  r22× t21   +  r23×t31

Two Matrices that can not be multiplied

Matrix C and D below cannot be multiplied together because the number of columns in C does not equal the number of rows in D. In this case, the multiplication of these two matrices is not defined.

Matrix Multiply

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

when multiplication is defined for matrices

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.

OK, so how do we multiply two matrices?

In order to multiply matrices,

  • Step 1: Make sure that the the number of columns in the 1st one equals the number of rows in the 2nd one. (The pre-requisite to be able to multiply)
  • Step 2: Multiply the elements of each row of the first matrix by the elements of each column in the second matrix.
  • Step 3: Add the products.

It's easier to understand if you go through the power point examples below.

A Step By Step Example

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Practice Problems

Practice 1
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Practice 2
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