Matrices formulas

Determinants of 2 × 2 and 3 × 3 matrices, the 2 × 2 inverse, and matrix multiplication.

These pages explain formulas for reference. They don't mean Mathover can solve every problem that uses them — interactive calculations are limited to the linked tools.

Determinant of a 2 × 2 matrix

det [[a, b], [c, d]] = ad − bc

Multiply down the main diagonal and subtract the product of the other diagonal.

Symbols

a, b, c, d: entries, row by row

When it applies

Square 2 × 2 matrices. det = 0 means the matrix has no inverse.

Worked example

Find det [[3, 2], [1, 4]].

1.ad = 3 × 4 = 12
2.bc = 2 × 1 = 2
3.12 − 2 = 10

Answer: 10

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Determinant of a 3 × 3 matrix

det = a(ei − fh) − b(di − fg) + c(dh − eg)

Expand along the top row: each entry times the 2 × 2 determinant left when you cover its row and column, with signs + − +.

Symbols

a … i: entries of [[a, b, c], [d, e, f], [g, h, i]]

When it applies

Square 3 × 3 matrices.

Worked example

Find det [[2, 0, 1], [1, 3, 2], [1, 1, 2]].

1.2 × (3·2 − 2·1) = 2 × 4 = 8
2.− 0 × (1·2 − 2·1) = 0
3.+ 1 × (1·1 − 3·1) = −2
4.8 − 0 − 2 = 6

Answer: 6

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Inverse of a 2 × 2 matrix

[[a, b], [c, d]]⁻¹ = (1 ÷ (ad − bc)) × [[d, −b], [−c, a]]

Swap a and d, change the signs of b and c, then divide every entry by the determinant.

Symbols

a, b, c, d: entries
ad − bc: the determinant

When it applies

Only when ad − bc ≠ 0. A matrix with determinant 0 has no inverse.

Worked example

Find the inverse of [[3, 2], [1, 4]].

1.det = 12 − 2 = 10
2.Swap and negate: [[4, −2], [−1, 3]]
3.Divide by 10: [[2/5, −1/5], [−1/10, 3/10]]

Answer: [[2/5, −1/5], [−1/10, 3/10]]

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Matrix multiplication

(AB)ᵢⱼ = Σₖ Aᵢₖ × Bₖⱼ

Each entry of the product is row i of A multiplied, entry by entry, with column j of B, then added.

Symbols

A: an m × n matrix
B: an n × p matrix
AB: the m × p product

When it applies

The number of columns of A must equal the number of rows of B. In general AB ≠ BA.

Worked example

Multiply [[1, 2], [3, 4]] by [[5, 6], [7, 8]].

1.Row 1: 1·5 + 2·7 = 19, 1·6 + 2·8 = 22
2.Row 2: 3·5 + 4·7 = 43, 3·6 + 4·8 = 50

Answer: [[19, 22], [43, 50]]

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