Linear Algebra · Quick Reference

Determinants Cheat Sheet

Linear Algebra · Lesson 3/7

In one line: the determinant condenses an entire square matrix into a single number that reveals whether the matrix is invertible and how it scales area or volume.

Key Ideas

1Determinant of a 2x2 Matrix. For [[a,b],[c,d]], det = a*d - b*c.

2Determinant of a 3x3 Matrix (Cofactor Expansion). Expand along the first row, multiplying each entry by the determinant of the 2x2 matrix formed by removing its row and column, alternating signs (+,-,+).

3Determinant and Invertibility. A square matrix is invertible if and only if its determinant is nonzero.

4Determinant and Scaling. The absolute value of the determinant tells you how much a matrix scales area (in 2D) or volume (in 3D) when used as a transformation.

5Properties of Determinants. Swapping two rows flips the sign of the determinant. A matrix with a row of all zeros has determinant 0. det(A*B) = det(A)*det(B).

Worked Examples

Find the determinant of [[3,1],[2,4]].

det = 10

Find the determinant of [[1,2,3],[0,1,4],[5,6,0]] using cofactor expansion along the first row.

det = 1

If det(A)=3 and det(B)=4, find det(A*B).

det(A*B) = 12

Formulas

det([[a,b],[c,d]]) = ad - bc

det(A*B) = det(A) * det(B)

Practice Yourself

Find the determinant of [[2,3],[1,4]].

Find the determinant of [[6,2],[3,1]].

Is a matrix with determinant 0 invertible?