Concepts · 3 min read
Matrices for Beginners: Multiplication, Determinant, Inverse
Multiplying matrices
Each entry of the product is a row of the first matrix times a column of the second (multiply pairs, then add).
| Entry | Working | Value |
|---|---|---|
| (1,1) | 1×5 + 2×7 | 19 |
| (1,2) | 1×6 + 2×8 | 22 |
| (2,1) | 3×5 + 4×7 | 43 |
| (2,2) | 3×6 + 4×8 | 50 |
Order matters: A × B is usually not B × A.
Determinant of a 2×2
For [[a,b],[c,d]] the determinant is ad − bc. Here 1×4 − 2×3 = -2. A non-zero determinant means the matrix can be inverted.
The inverse
A⁻¹ = (1/det) × [[d, −b], [−c, a]] = [[−2, 1], [1.5, −0.5]]. Multiplying A by A⁻¹ gives the identity matrix.
Where matrices are used
They describe transformations in computer graphics, solve systems of equations and model networks. For single equations see the linear equation solver.
Frequently asked questions
Can every matrix be inverted?
Only square matrices with a non-zero determinant.
Why is matrix multiplication not commutative?
Because rows of A pair with columns of B; swapping the order pairs different numbers.
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