Matrix Multiplication: 2x3 times 3x2

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Find the product AB = C, where A =

┌				┐
│	4	1	5	│
│	3	–2	1	│
└				┘

and B

┌			┐
│	1	–4	│
│	3	2	│
│	–2	–3	│
└			┘

We’ll be talking about rows and columns, so let’s be sure which is which. Is a row left-right or up-and-down?

Left-right.

These matrices have different numbers of rows and columns. Can they really be multiplied?

Yes they can.

What is the requirement on the number of rows and columns of A and B?

The only requirement is that the number of columns in the first matrix must be the same as the number of rows in the second.

Will the result of AB = C be a number or a matrix?

A matrix.

How many rows will C have?

It will have the same number of rows as the first matrix has.

How many is that here?

Two.

How many columns will C have?

It will have the same number of columns as the second matrix has.

How many is that here?

Two.

Use C₁₁, C₁₂, C₂₁, and C₂₂ to represent the elements of C. Show how C will look.

┌			┐
│	C₁₁	C₁₂	│
│	C₂₁	C₂₂	│
└			┘

Now we can set up our problem:

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

│

–2

│

C₂₁

C₂₂

│

└

┘

│

–2

–3

│

└

┘

└

┘

The color indicates how we obtain C₁₁. Show the details with the numerical values.

We get C₁₁ = 4*1 + 1*3 + 5*(–2) = –3

Which row of A is involved in C₁₂?

The first.

Which column of B is involved in C₁₂?

The second.

Indicate this on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

│

–2

│

C₂₁

C₂₂

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

We get C₁₂ = 4*(–4) + 1*2 + 5*(–3) = –29

Which row of A is involved in C₂₁?

The second.

Which column of B is involved in C₂₁?

The first.

Indicate this on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

│

–2

│

C₂₁

C₂₂

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

We get C₂₁ = 3*1 + (–2)*3 + 1*(–2) = –5

Which row of A is involved in C₂₂?

The second.

Which column of B is involved in C₂₂?

The second.

Indicate this on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

│

–2

│

C₂₁

C₂₂

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

We get C₂₂ = 3*(–4) + (–2)*2 + 1*(–3) = –19

Combine the results to show the numerical elements of C.

		┌			┐		┌			┐
		│	C₁₁	C₁₂	│		│	–3	–29	│
C	=	│	C₂₁	C₂₂	│	=	│	–5	–19	│
		└			┘		└			┘

Next, let’s try forming the product

.
What determines whether the product can be found?

The number of columns in the first matrix must equal the number of rows in the second matrix.

Is that the case here?

Yes.

How many rows will the product have?

It will have the same number of rows as the first factor has.

How many columns will the product have?

It will have the same number of columns as the second factor has.

State the dimensions of our product.

It will be 3 rows by 3 columns.

Set up the first row of the product.

Set up the second row of the product.

Set up the third row.

Simplify.

Obviously,

when the dimensions of the two products are different.

Will the dimensions of the two products ever be equal?

Yes, when the two matrices are square.

Here is a 2x2 example:

, and

Determine the product AB.

We get

Determine BA.

So, even though the dimensions are the same for the two products, they are not equal. This is usually the case, but not always.

The end. If you found this helpful and would recommend that I create more pages like this one, please let me know: Email to John Taylor

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