Unit 9 · Advanced TopicsN5 · Matrices · Propulsion

Matrices

Add entries, multiply rows by columns, check dimensions, and use determinants.

28 min
Unlocks: Gimbal Array

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

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Warm-up

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Key terms

matrix: A rectangular array of entries.
dimension: Rows by columns.
determinant: For a 2 by 2 matrix, ad-bc.

Start by naming the quantity each problem asks for. That habit blocks the most common ACT trap.

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Try it first

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Take a calm first shot. Label what you know, write one relationship, and make the best choice you can.

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The big idea

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Build the picture before the arithmetic

Addition is entry by entry and requires equal dimensions. For AB, columns of A must equal rows of B. Each output entry is a dot product, not an elementwise product.

Key idea

The decision that matters

Matrix multiplication is row by column. The inside dimensions must match; the outside dimensions name the product.

2 by 2 determinant

\det\begin{pmatrix}a&b\\c&d\end{pmatrix}=ad-bc

Write this relationship before inserting numbers. It keeps labels and units attached to the work.

A compact example

$\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}5\\6\end{pmatrix}=\begin{pmatrix}17\\39\end{pmatrix}$ .

Cue	Best move
Can AB exist?	inside dimensions must match
Find an entry	take a row-column dot product

Trap

Multiplying corresponding entries

This error produces a believable answer. Pause before computing and say what each number means.

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Worked examples

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Skip check · 2 questions

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Worked example

Example 1: Multiply row by column

Problem

Compute

\begin{bmatrix}2&-1\\3&4\end{bmatrix}\begin{bmatrix}5\\2\end{bmatrix}

First entry:

2(5)+(-1)(2)=8

Second entry: $3(5)+4(2)=23$ .

How is each entry of the matrix product computed?

The product is $\begin{bmatrix}8\\23\end{bmatrix}$ .

Why does each row produce one output entry?

Worked example

Example 2: Solve with a determinant

Problem

For

M=\begin{bmatrix}k&3\\2&5\end{bmatrix}

, find

k

\det(M)=14

The determinant is

5k-3(2)=5k-6

Set $5k-6=14$ .

Which determinant equation represents the condition?

Then $5k=20$ , so $k=4$ .

Why is the determinant 5k minus 6?

Faded example · you finish it

Example 3: You finish it

Problem

2\times3

matrix multiplies a

3\times4

matrix. Complete the product dimension.

Check that the inside dimensions match; the outside dimensions name the product.

Which completion is correct?

Which reason validates the result?

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Practice

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Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

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Exit ticket

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Optional · 4 questions · Hardest levelChallenge set: the kind of question that decides a 36

These are last-ten-questions hard. Take your time, use Desmos when it helps, and expect to miss some: that's where the learning is.