Matrices

Higher Level

Each entry of the product comes from one row meeting one column. Step through the four entries and watch which numbers are used.

row 1, col 1

1×0 + 2×1the arithmetic

2that entry

Row across, column down, multiply and add.

Order matters

A= [[1, 2], [3, 4]] B= [[0, 1], [1, 0]] AB= [[2, 1], [4, 3]] BA= [[3, 4], [1, 2]] not the same

Matrix multiplication does not commute. Every other product you have met does, so this has to be learned rather than assumed. It also means you cannot "divide both sides" without deciding which side you are multiplying on.

And it is not elementwise. Multiplying the top-left entries would give 1 × 0 = 0. The real answer is 2, because it uses the whole first row and the whole first column.

Orders, and when a product exists

An m × n matrix times an n × p matrix gives an m × p matrix. The inside numbers must match, and the outside ones give the answer's size. A 2×3 times a 3×2 works; a 2×3 times a 2×3 does not exist at all.

The easy operations

add→ entry by entry, same order only scalar→ multiply every entry: 2A = [[2, 4], [6, 8]] equal→ same order and every entry matching det of A= 1(4) − 2(3) = −2

Worked example

A school shop buys shirts at 320 baht and ties at 180. Branch A orders 45 shirts and 30 ties; branch B orders 60 shirts and 12 ties.

Use matrix multiplication to find what each branch spends, and state the order of every matrix involved.

Orders O = [[45, 30], [60, 12]], a 2 × 2Rows are branches, columns are items. Deciding this before multiplying is the whole job.
Prices P = [[320], [180]], a 2 × 1A column, because the prices must line up with the columns of O.
O × P is (2×2)(2×1), so the inner 2s match and the result is 2 × 1Check this first. A 1 × 2 price row would not multiply at all in this order.
Row 1: 45(320) + 30(180) = 14400 + 5400Row across, column down, multiply and add.
= [[19800], [21360]]: branch A spends 19,800 baht, branch B 21,360Row 2 is 60(320) + 12(180) = 21,360. Note branch B ordered fewer items and spent more, which the matrix gives you without any separate reasoning.

Your turn

1. With A and B above, find the top-left entry of AB.

2. Find the top-left entry of BA.

3. A 2×3 matrix multiplied by a 3×2 matrix gives:

A 2×2 matrix. A 3×3 matrix. A 2×3 matrix. Nothing; it is not defined.

Where the marks go

Checking the orders are compatible before multiplying anything.

Keeping the order of the product as the question states it. AB and BA are different answers and only one of them is being asked for.

Interpreting the result in context. A matrix of costs times a matrix of quantities gives totals, and saying which is which earns the mark.

Come back to this

These three are deliberately not all about this page. In an examination the hardest step is deciding which method applies, and questions that arrive straight after the method never make you decide. Answer from memory before you reveal anything.

Come back to these in a week, and again in a month.

Which method?

M = [[2,1],[1,2]]. Find its eigenvalues.

Characteristic polynomial: λ² − 4λ + 3 = 0, so 3 and 1. They sum to the trace and multiply to the determinant, which checks the work.
Which method?

Write 52 × 10⁶ in standard form.

Standard form: 5.2 × 10⁷. a must be at least 1 and under 10, so 52 is not finished.
Which method?

Is AB always equal to BA?

No. Matrix multiplication does not commute, which is unlike every other product you have met and is why you cannot simply divide both sides.

The same idea elsewhere

4.19, transition matrices: A transition matrix is a matrix, and the chain is repeated multiplication.
1.8, solving with technology: A linear system is a matrix equation in disguise.

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