Matrix Multiplication: 2x3 times 3x3

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Find the product AB = C, where A =

┌				┐
│	4	1	5	│
│	3	–2	1	│
└				┘

and B

┌				┐
│	1	–4	5	│
│	3	2	0	│
│	–2	–3	2	│
└				┘

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?

Three.

Use C₁₁, C₁₂, C₁₃, C₂₁, C₂₂, and C₂₃ to represent the elements of C. Show how C will look.

┌				┐
│	C₁₁	C₁₂	C₁₃	│
│	C₂₁	C₂₂	C₂₃	│
└				┘

Now we can set up our problem:

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–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 with highlighting on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–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 C₁₃?

The first.

Which column of B is involved in C₁₃?

The third.

Indicate this with highlighting on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

We get C₁₃ = 4*5 + 1*0 + 5*2 = 30

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

–5

│

C₁₁

C₁₂

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–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

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

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

Which row of A is involved in C₂₃?

The second.

Which column of B is involved in C₂₃?

The third.

Indicate this on the matrices.

┌

┐

┌

┐

┌

┐

│

–4

│

C₁₁

C₁₂

C₁₃

│

–2

│

C₂₁

C₂₂

C₂₃

│

└

┘

│

–2

–3

│

└

┘

└

┘

Show the details with the numerical values.

We get C₂₃ = 3*5 + (–2)*0 + 1*2 = 17

Combine the results to show the numerical elements of C.

┌

┐

┌

┐

│

C₁₁

C₁₂

C₁₃

│

–3

–29

│

C₂₁

C₂₂

C₂₃

│

–5

–19

│

└

┘

└

┘

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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