Calculating Determinants

Algebra 2 · Axiom Academy

LESSON Calculating Determinants From the rule to cofactor expansion — and what a determinant reveals about a matrix. Every square matrix has a determinant : one number that encodes key geometric and algebraic information. Its columns are two vectors, and the determinant is the (signed) area of the parallelogram they span — how much the matrix scales space. A determinant can be positive, negative, or zero: its sign records orientation, and its size records how much area (or volume) the matrix scales by. For a matrix the determinant is the product along the main diagonal minus the product along the anti-diagonal . Example: evaluate a determinant For a matrix we use cofactor expansion (Laplace expansion) along any row or column — typically the first row. Each entry multiplies the determinant of its minor : what remains after deleting that entry's row and column. Watch the three signed cofactor terms accumulate on the number line — the running total lands exactly on the determinant. 5. Determinants and Invertibility The determinant answers a crucial question: can the matrix be undone? Watch a parallelogram collapse as its columns line up — its area, the determinant, shrinks to zero. If , the matrix has an inverse. If , it is singular — no inverse exists. Since , matrix A is invertible. Since , matrix B is not invertible — its columns are parallel. You can now compute and determinants — and read what the value says about a matrix. Scroll up to revisit any step.

This is the written version of the interactive lesson above. See the full Algebra 2 course.