Mathematics
Matrices and Determinants
A matrix is a rectangular grid of numbers; a determinant is a single scalar derived from a square matrix. Matrices encode linear transformations; determinants tell you whether the transformation is invertible (det ≠ 0) and how much it scales area or volume.
How to Approach Matrices and Determinants
Match dimensions for operations
Add/subtract: matrices must be the same size. Multiply: A·B requires A's columns to equal B's rows. The result's dimensions are A's rows × B's columns.
Compute the determinant
For 2×2: ad - bc. For 3×3: expand along a row or column using cofactors. For larger: use row reduction to upper triangular form, then multiply the diagonal.
Find the inverse (if det ≠ 0)
A·A⁻¹ = I. For 2×2: A⁻¹ = (1/det)·[[d,-b],[-c,a]]. For larger: use Gauss-Jordan elimination on [A | I] until you get [I | A⁻¹].
Frequently Asked Questions
Why does matrix multiplication not commute?+
Because matrices represent linear transformations, and the order of transformations matters geometrically. Rotating then reflecting ≠ reflecting then rotating.
What does the determinant represent?+
The absolute value of the determinant is the factor by which the matrix scales area (2D) or volume (3D). A negative determinant means the orientation is flipped.
When does a matrix have no inverse?+
When the determinant is zero. That happens when the columns (or rows) are linearly dependent — they collapse the transformation to a lower dimension.
Related Topics
More step-by-step guides in Mathematics and adjacent subjects.
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