VG3
linear-algebra
Quiz
What are determinants?
The determinant is a single number that tells us important properties of a matrix — including whether it can be inverted.
The determinant is a number associated with a square matrix. It tells us whether the matrix has an inverse, and measures how the matrix scales areas and volumes.
Determinant of a 2×2 matrix
det(A) = |a b| = ad - bc
|c d|
Example:
A = [3 2], [1 4]
det(A) = 3·4 - 2·1 = 10
A = [3 2], [1 4]
det(A) = 3·4 - 2·1 = 10
What does the determinant tell us?
det(A) ≠ 0 → A is invertible (unique solution)
det(A) = 0 → A is singular (no unique solution)
det(A) = 0 → A is singular (no unique solution)
Geometric interpretation:
|det(A)| = the area of the parallelogram formed by the column vectors.
If det(A) = 6, the matrix scales areas by factor 6.
|det(A)| = the area of the parallelogram formed by the column vectors.
If det(A) = 6, the matrix scales areas by factor 6.
The determinant reveals the inner nature of a matrix — like a fingerprint of its essence.
— Gottfried Wilhelm Leibniz (1646–1716)
🧠 Test yourself
Question 1 of 5
What is det([3 1; 2 4])?