4.8 Matrices

What is a matrix?

A matrix is a rectangular array of numbers organized into rows and columns, described by its dimensions (rows × columns). Matrices can represent systems of equations, transformations, or data sets. Entries are located by row and column position, like a coordinate grid.

Adding, subtracting, and scalar multiplying

Matrices of the same dimensions can be added or subtracted by combining corresponding entries. Scalar multiplication multiplies every entry by a constant. These operations follow rules similar to arithmetic with real numbers, but matrix size must match for addition/subtraction.

Matrix multiplication

To multiply matrix A (m×n) by matrix B (n×p), the number of columns in A must equal the number of rows in B. Each entry in the product is found by taking the dot product of a row from A and a column from B. Matrix multiplication is not commutative: AB ≠ BA generally.

Key terms

Matrix

A rectangular array of numbers arranged in rows and columns.

Matrix dimensions

Described as rows × columns, e.g., a 2×3 matrix has 2 rows and 3 columns.

Condition for matrix multiplication

The number of columns in the first matrix must equal the number of rows in the second matrix.

Scalar multiplication

Multiplying every entry in a matrix by the same constant, keeping dimensions unchanged.

Is matrix multiplication commutative?

No — generally AB ≠ BA, since order affects the resulting product.

Try an AP-style question

[[1, 2], [3, 4]] + [[0, 1], [1, 0]] = ?

1. [[2, 3], [4, 5]]
2. [[0, 2], [3, 0]]
3. [[1, 3], [4, 4]]
4. [[1, 2], [3, 4]]