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Matrices

You’ll be able to

Matrices organize numbers in a grid

A matrix is a rectangular array of numbers arranged in rows and columns. Its dimensions are written rows × columns, so a 2×3 matrix has 2 rows and 3 columns. Matrices store systems of equations, coordinate data, and — as the next lesson shows — the recipes for geometric transformations.

The 2×2 determinant and matrix products

For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc, a single number that measures how the matrix scales area (and whether it is invertible). To multiply matrices, the inner dimensions must match: an (m×n) times an (n×p) is allowed and yields an m×p result. So a 2×3 times a 3×2 gives a 2×2 matrix.

Determinant and product size
det[[a, b], [c, d]] = ad − bc · (m×n)(n×p) = m×p
The determinant is the main diagonal product minus the anti-diagonal product. Matrix multiplication requires the first matrix’s column count to equal the second’s row count.
Worked example

Find the determinant of the matrix [[2, 1], [3, 4]].

  1. 1.Identify the entries: a = 2, b = 1, c = 3, d = 4.
  2. 2.Multiply the main diagonal: ad = 2·4 = 8.
  3. 3.Multiply the anti-diagonal: bc = 1·3 = 3.
  4. 4.Subtract: det = ad − bc = 8 − 3 = 5.
Answer: The determinant is 5 (from 2·4 − 1·3 = 8 − 3). Since it is nonzero, the matrix is invertible.
Checkpoint

What is the determinant of [[2, 1], [3, 4]]?

Watch out

The determinant subtracts the anti-diagonal product: ad − bc. Adding the two products, or forgetting the second one, is the most common slip. A zero determinant means the matrix has no inverse.

Checkpoint

What are the dimensions of the product of a 2×3 matrix and a 3×2 matrix?

On the exam

Before multiplying, write the dimensions side by side, (2×3)(3×2), and check that the inner numbers match. The outer numbers give the answer’s size — a fast way to catch an impossible product on the exam.

Answer the 2 checkpoints as you read.

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