Determinants
Objective. Compute and interpret determinants algebraically and geometrically.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The determinant of a square matrix is a scalar encoding whether the matrix is invertible (det≠0) and, geometrically, the signed scale factor by which the matrix transforms area (2×2) or volume (3×3). For a 2×2 matrix [[a,b],[c,d]], det=ad−bc; for 3×3 matrices, cofactor expansion along a row or column reduces the computation to a sum of signed 2×2 determinants. For A=[[1,2],[3,4]], det(A)=(1)(4)−(2)(3)=−2, and the negative sign indicates the transformation reverses orientation. Determinants also satisfy det(AB)=det(A)det(B), a multiplicative property useful for composed transformations.
Worked example
Compute the determinant of the matrix A=[[2,3],[1,4]].
- 1Identify entries: a=2, b=3, c=1, d=4.
- 2Apply the 2×2 determinant formula det(A)=ad−bc.
- 3Compute ad=2×4=8 and bc=3×1=3.
- 4Subtract: det(A)=8−3=5.
Answer: 5
Concept mastery
Based on your past attempts at this lesson — the question types to practise again come first.
Complete a practice set to see your concept breakdown here.
Practice
Practice happens on its own screen, one question at a time. Answers stay hidden until you submit yours, and you can stop and pick up at the same question later.
Start practisingCommon mistake. Students expanding a 3×3 determinant by cofactors forget to alternate the signs (+,−,+) across the row or column, applying all positive signs and producing an incorrect value.
Your progress
Tick each step as you finish it. Your exact reading position is saved automatically, so Continue drops you back on the same line.
0 of 5 steps complete
Your place saves automatically as you read.