Matrix Multiplication

Matrix multiplication repeats the matrix-vector idea.

If $A$ multiplies a matrix $B$ , each column of $B$ acts like an input vector. The output is a new matrix whose entries come from row-column dot products.

Order matters. In $AB$ , rows come from $A$ and columns come from $B$ . Swapping the order changes the row-column pairings, and sometimes the swapped product is not even defined.

Let:

A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}

and:

B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}

The first entry of $AB$ is row 1 of $A$ dotted with column 1 of $B$ :

(1 \cdot 5) + (2 \cdot 7) = 19

The top-right entry uses row 1 of $A$ and column 2 of $B$ :

(1 \cdot 6) + (2 \cdot 8) = 22

The full product is:

AB = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}

The diagram highlights the top-right entry: row 1 of $A$ with column 2 of $B$ .

One entry of a matrix product comes from one row and one column.

MATH-C03-T05-001Exercise: Compute one entry

Using the matrices above, what is the top-right entry of $AB$ ?

Your answer

Compute it first, then check your number.

HintRow and column

Use row 1 of $A$ and column 2 of $B$ .

SolutionTop-right entry

(1 \cdot 6) + (2 \cdot 8) = 6 + 16 = 22

Top-right means first row of $A$ and second column of $B$ . Matrix multiplication fills each output entry this way.

MATH-C03-T05-005Exercise: Check product order

Let $A$ have shape 2 x 3 and $B$ have shape 3 x 4.

Enter 1 if $AB$ is defined but $BA$ is not defined.

Your answer

Compute it first, then check your number.

HintWrite both shapes

Compare (2 x 3)(3 x 4) with (3 x 4)(2 x 3).

SolutionOrder changes the middle dimensions

$AB$ has shape:

(2 x 3)(3 x 4)

Here the middle dimensions match. But $BA$ would be:

(3 x 4)(2 x 3)

In $BA$ , the middle dimensions are 4 and 2. They do not match, so $BA$ is not defined.

Shape Rule

If:

A has shape m x n
B has shape n x p

then:

AB has shape m x p

The middle dimensions must match.

The result keeps the outside dimensions: rows from $A$ , columns from $B$ .

MATH-C03-T05-002Exercise: Predict product shape

If $A$ has shape 3 x 2 and $B$ has shape 2 x 4, what is the shape of $AB$ ?

Your answer

Compute it first, then check your number.

HintKeep the outside

(3 x 2)(2 x 4) keeps 3 and 4.

SolutionOutside dimensions

The middle dimensions are both 2, so the product is allowed:

(3 \times 2)(2 \times 4) \rightarrow 3 \times 4

The output has 3 rows from $A$ and 4 columns from $B$ .

When Shapes Do Not Fit

Matrix multiplication is not always allowed.

If $A$ has shape 3 x 2 and $B$ has shape 4 x 2, then $AB$ is not defined. The 2 columns of $A$ cannot pair with the 4 rows of $B$ .

MATH-C03-T05-003Exercise: Spot invalid multiplication

Enter 1 if (3 x 2)(4 x 2) is a valid matrix product, or 0 if it is not.

Your answer

Compute it first, then check your number.

HintCheck middle dimensions

The middle dimensions are the second number of the first shape and the first number of the second shape.

SolutionMiddle mismatch

Enter 0. The product would require:

(3 x 2)(4 x 2)
     ^  ^

The middle dimensions are 2 and 4. They do not match, so the product is not defined.

Not Coordinate-Wise Multiplication

Matrix multiplication is not entry-by-entry multiplication.

For the matrices above, the top-left entry of $AB$ is 19. It is not:

1 x 5 = 5

It is:

(1 \cdot 5) + (2 \cdot 7) = 19

MATH-C03-T05-004Exercise: Avoid entry-wise multiplication

Enter 1 if matrix multiplication uses row-column dot products, not simple coordinate-wise multiplication.

Your answer

Compute it first, then check your number.

HintOne entry uses many numbers

The top-left entry used both 1 and 2 from the first row of $A$ .

SolutionDot products, not matching cells

Enter 1. Each entry of $AB$ comes from a row of $A$ dotted with a column of $B$ .

Code Mirror

In NumPy, @ performs matrix multiplication:

import numpy as np

A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])

print(A @ B)

The output is:

[[19 22]
 [43 50]]

Next, we name three matrix patterns that appear often.