Properties of Determinants

Linear Algebra (Matrices) · Axiom Academy

LESSON Row Operations and the Determinant Three row moves, three effects on — the trick that lets you reduce any matrix to triangular form and read its determinant off the diagonal. Every row operation changes the determinant in one fixed, predictable way. Watch the same matrix below — its determinant readout reacts the instant each operation is applied. Scale one row by k : scales by k Add a multiple of one row to another: no change 2. The Payoff: Reduce, Then Read the Diagonal Here is why those three rules matter. Use them to drive any matrix to triangular form. Add-a-multiple moves do the eliminating for free; a swap (when you need one) just flips the sign. Once the matrix is triangular, the determinant is simply the product of the diagonal — corrected by whatever sign the swaps cost. Reading the animation: one swap put the sign at -1 , then two add-a-multiple moves cleared the first column and the (3,2) entry without changing anything. The matrix is now upper-triangular. The same rules tell you when a determinant must be 0 without any computation . If one row is a multiple of another, a single add-a-multiple move — which never changes — turns one of them into a row of zeros. A zero row forces , so the original determinant was 0 all along. Watch it happen. That argument generalizes. A few more facts fall straight out of the three rules: — there is nothing to scale a diagonal product against. — they reduce to a zero row, exactly as in the animation.

This is the written version of the interactive lesson above. See the full Linear Algebra (Matrices) course.