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How to Multiply Matrices in Math

Updated on March 28, 2013

What is a Matrix?

A Matrix is a set of numbers. More specifically a rectangular array of elements. A matrix (one) or matrices (multiple) are organized in rows and columns surrounded by box brackets or parenthesis. Matrices have many purposes in higher level math and physics. Linear systems of equations, transformations, and linear combinations are just some ways in which matrices are used.

These elements can be numbers or variables. We can also notate the variable with a variable with two subscripts separated by a comma to notate the place that it is in. This is common notation in textbooks but can be extremely confusing for most students learning about this concept.

Multiplying Matrices Sizes

There is one very important rule about multiplying matrices. The number of columns in the first matrix must match the number of row in the second matrix.

row by column

2by3 x 3by4 YES!

2by3 x 4by3 NO!

We are going to start with an example that has a 2 by 2 matrix times a 2 by 1 matrix.

The first matrix has 2 columns and the second one has 2 rows, we are able to multiple these matrices.

The new size of the multiplied matrix is the first matrix's row by the second matrix's columns. So the new size is a 2 by 1.


Steps to Multiply Matrices

We multiply matrices go left to right across the first matrix and top to bottom on the second matrix.

There will be two rows in the final matrix.

We multiply one element time one element and add the products together. In this example let's look at one row at a time.

The first row is 1 time 5 plus 2 times 6

5 plus 12

Equals 17

The second row is 3 times 5 plus 4 times 6.

15 plus 24

Equals 39


Multiply 2 by 2 Matrix

This 2 by 2 matrix multiplication will start being a 2 by 2 and end being a 2 by 2. Let's see how this happens.

Remember left to right and top to bottom

Row 1 by column 1, Row 1 by column 2

Row 2 by column 1, Row 2 by column 2

1 times 5 plus 2 times 7 , 1 times 6 plus 2 times 8

3 times 5 plus 4 times 7, 3 times 6 plus 4 times 8

Multiplying Two Matrices

The last examples I am going to show is a 2 by 3 and a 3 by 1.

Remember the rule across left to right in the first matrix and top to bottom in the second matrix.

This matrix ends up being a 2 by 1 matrix.

Many of the term combine in these two matrices in order to create the final matrix.

More Multiplying Matrices Examples

Matrix Multiplication Limits

Matrix multiplication is not commutative, meaning you can not switch order in which you multiply the matrices in.

There is an identity matrix for multiplication when the diagonal of the matrix is 1 going from the top left of the bottom right, while the rest of the elements are zero.

Try Multiplying Matrices

These three different examples show you how to multiply different sizes and allow you to see how the multiplication of matrices work.


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    • Caleb DRC profile image

      Caleb DRC 4 years ago

      Very well done and thorough.