Properties of Determinants

Key idea

Multiplicative; row swaps change sign.

In depth

det(AB) = det A det B, det Aᵀ = det A, swapping two rows changes the sign, multiplying a row by k multiplies det by k, and adding a multiple of one row to another changes nothing. Triangular matrices have det equal to the product of the diagonal.

Worked example

det(2A) for a 3 × 3 matrix equals 2³ det A = 8 det A.

Formula

Real-world uses

• Checking the effect of row operations on a system
• Testing if a matrix is invertible

Common mistake

Assuming det(A + B) = det A + det B.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Linear Algebra

• Triple Product – Scalar triple product gives parallelepiped volume.
• Homogeneous Systems – Ax = 0 always has the trivial solution.
• Cofactor Expansion – Compute determinants recursively.
• Geometric Meaning of Determinants – Area or volume scaling factor.
• Cramer’s Rule for Systems – Solve systems using determinants.
• Adjugate Matrix – Transpose of the cofactor matrix.