01 Key Concepts
Determinant of a 2x2 Matrix
For [[a,b],[c,d]], det = a*d - b*c.
Determinant 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 (+,-,+).
Determinant and Invertibility
A square matrix is invertible if and only if its determinant is nonzero.
Determinant 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.
Properties 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).
02 Key Formulas
- det([[a,b],[c,d]]) = ad - bc
- det(A*B) = det(A) * det(B)
03 Solved Examples
Example 1 Find the determinant of [[3,1],[2,4]].
- Apply the 2x2 formula: (3*4) - (1*2) = 12-2.
Answer: det = 10
Example 2 Find the determinant of [[1,2,3],[0,1,4],[5,6,0]] using cofactor expansion along the first row.
- Expand: 1*det([[1,4],[6,0]]) - 2*det([[0,4],[5,0]]) + 3*det([[0,1],[5,6]]).
- det([[1,4],[6,0]]) = 0-24 = -24. det([[0,4],[5,0]]) = 0-20 = -20. det([[0,1],[5,6]]) = 0-5 = -5.
- 1*(-24) - 2*(-20) + 3*(-5) = -24+40-15.
Answer: det = 1
Example 3 If det(A)=3 and det(B)=4, find det(A*B).
- Apply the property det(A*B) = det(A)*det(B).
- 3*4.
Answer: det(A*B) = 12
04 Practice Questions
1Find the determinant of [[2,3],[1,4]].
5
2Find the determinant of [[6,2],[3,1]].
0
3Is a matrix with determinant 0 invertible?
No
4If det(A)=5 and det(B)=2, find det(A*B).
10
5What happens to the determinant if you swap two rows of a matrix?
It flips sign (multiplies by -1)
📄 Determinants — Downloadable Worksheet
10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.