Algebra · Quick Reference

Determinants Cheat Sheet

Algebra II · Lesson 16/17

In one line: the determinant is a single number calculated from a square matrix that reveals important properties, like whether the matrix can be inverted at all.

Key Ideas

1Determinant of a 2x2 Matrix. For [[a,b],[c,d]], the determinant equals a*d - b*c.

2Notation. The determinant of matrix A is written det(A) or with vertical bars around the matrix entries.

3What the Determinant Tells You. If det(A) = 0, the matrix is 'singular' and has no inverse. If det(A) is not 0, the matrix does have an inverse.

4Determinant of a 3x3 Matrix. Calculated using cofactor expansion, breaking the 3x3 determinant down into a combination of three 2x2 determinants.

5Determinants and Area. For a 2x2 matrix formed from two vectors, the absolute value of the determinant equals the area of the parallelogram those vectors form.

Worked Examples

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

det = 10

Find the determinant of [[2,4],[1,2]] and determine if it's invertible.

det = 0, so this matrix is NOT invertible (it's singular)

Find the determinant of [[5,0],[0,5]].

det = 25

Formulas

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

Practice Yourself

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

-2

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

Is a matrix with determinant 0 invertible?