How to Add, Multiply & Find the Determinant of a Matrix
A matrix is a rectangular grid of numbers. Matrix operations underpin computer graphics, machine learning, physics simulations, and systems of linear equations. Addition is straightforward element-wise, while multiplication follows a row-times-column rule that requires the inner dimensions to match. The determinant tells you whether a matrix can be inverted.
Last updated: March 31, 2026
The Formula
Addition: [A + B]ᵢⱼ = Aᵢⱼ + Bᵢⱼ Multiplication: [A × B]ᵢⱼ = Σ(row i of A · col j of B) Determinant 2×2: det(A) = ad − bc Inverse 2×2: A⁻¹ = (1/det) × [[d, −b], [−c, a]]
Variable Definitions
| Symbol | Name | Description |
|---|---|---|
| A, B | Matrices | For multiplication, the number of columns in A must equal the number of rows in B |
| det(A) | Determinant | A scalar indicating whether the matrix is invertible (det ≠ 0) and the scale factor of the transformation |
| A⁻¹ | Inverse | The matrix such that A × A⁻¹ = I (identity). Only exists when det ≠ 0. |
Step-by-Step Example
For A = [[2, 3], [1, 4]], find det(A) and A⁻¹.
Given
Solution
- 1Identify a, b, c, d:
a=2, b=3, c=1, d=4 - 2Determinant:
det = (2×4) − (3×1) = 8 − 3 = 5 - 3Swap a↔d, negate b and c:
Adjugate = [[4, −3], [−1, 2]] - 4Divide by determinant:
A⁻¹ = (1/5) × [[4, −3], [−1, 2]] = [[0.8, −0.6], [−0.2, 0.4]] - 5Verify A × A⁻¹ = I:
[[2,3],[1,4]] × [[0.8,−0.6],[−0.2,0.4]] = [[1,0],[0,1]] ✓
det(A) = 5. A⁻¹ = [[0.8, −0.6], [−0.2, 0.4]].
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Common Mistakes to Avoid
Multiplying matrices where inner dimensions don't match — m×n can only multiply n×p.
Assuming A×B = B×A — matrix multiplication is generally not commutative.
Inverting a singular matrix (det = 0) — it has no inverse.
Confusing the adjugate step — swap a and d, negate b and c (do not swap b and c).