4. Multiplication of Matrices

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"Flash matrix interactive"
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Important: We can only multiply matrices if the number of columns in the first matrix is the same as the number of rows in the second matrix.

Examples

Multiplying a 2 × 3 matrix by a 3 × 4 matrix is possible and it gives a 2 × 4 matrix as the answer.

Multiplying a 7 × 1 matrix by a 1 × 2 matrix is okay; it gives a 7 × 2 matrix

A 4 × 3 matrix times a 2 × 3 matrix is NOT possible.

How to Multiply 2 Matrices

We use letters first to see what is going on. We'll see a numbers example after.

As an example, let's take a general 2 × 3 matrix multiplied by a 3 × 2 matrix.

The answer will be a 2 × 2 matrix.

We multiply and add the elements as follows. We work across the 1st row of the first matrix, multiplying down the 1st column of the second matrix, element by element. We add the resulting products. Our answer goes in position a₁₁ (top left) of the answer matrix.

first

We do a similar process for the 1st row of the first matrix and the 2nd column of the second matrix. The result is placed in position a₁₂.

third

Now for the 2nd row of the first matrix and the 1st column of the second matrix. The result is placed in position a₂₁.

second

Finally, we do the 2nd row of the first matrix and the 2nd column of the second matrix. The result is placed in position a₂₂.

fourth

So the result of multiplying our 2 matrices is as follows:

2x3 by 3x2

Now let's see a number example.

Example

Multiply:

Answer

Multiplying 2 × 2 Matrices

The process is the same for any size matrix. We multiply across rows of the first matrix and down columns of the second matrix, element by element. We then add the products:

In this case, we multiply a 2 × 2 matrix by a 2 × 2 matrix and we get a 2 × 2 matrix as the result.

Example

Multiply:

Answer

Here is a (2×2)×(2×2) example in LiveMath.

LIVEMath

Let's look at another example. This time we have (3×3)×(3×1).

LIVEMath

Flash Interactive

Here's a Flash movie to play with. It will generate many different sized (up to 5 by 5) matrices with different random numbers each time. You can see plenty of examples of matrix operations, including adding, subtracting and multiplying.

You can step through each calculation involved. You can do this by clicking the "step" button which appears.

Suggestion: Work out the answer yourself first, then check your answer against what it says. Never just copy!

Loading Flash movie.

Matrices and Systems of Simultaneous Linear Equations

We now see how to write a system of linear equations using matrix multiplication.

Example:

The system of equations

can be written as:

Matrices are ideal for computer-driven solutions of problems because computers easily form arrays. We can leave out the algebraic symbols. A computer only requires the first and last matrices to solve the system, as we will see in Matrices and Linear Equations.