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Linear Algebra Unit 3: Determinants

The determinant is one number that tells you whether a matrix is invertible and how it scales area or volume.

Covers: Cofactor expansion, properties, Cramer's rule, geometric meaning

Study guide

The determinant is a number attached to a square matrix. For 2×2, det[a b; c d] = ad − bc. For larger matrices, cofactor expansion goes along any row or column: multiply each entry by its cofactor (a signed smaller determinant, with signs in a checkerboard pattern) and add. Expanding along a row or column with many zeros saves work. The determinant of a triangular matrix is the product of its diagonal entries.

Row operations affect the determinant predictably: swapping two rows changes its sign, multiplying a row by k multiplies it by k, and adding a multiple of one row to another doesn't change it. So row reducing to triangular form is an efficient way to compute determinants.

Key properties: det(AB) = det A·det B, det(Aᵀ) = det A, det(A⁻¹) = 1/det A, and det(cA) = cⁿ det A for an n×n matrix. A is invertible exactly when det A ≠ 0; a zero determinant means the matrix is singular.

Cramer's rule solves Ax = b when det A ≠ 0: xᵢ = det Aᵢ/det A, where Aᵢ is A with column i replaced by b. It's elegant for small systems but slow for large ones.

Geometrically, |det A| is the factor by which the transformation x ↦ Ax scales area (2D) or volume (3D), and a negative determinant means orientation is reversed.

The inverse can be written with cofactors as A⁻¹ = (1/det A)·Cᵀ, where C is the matrix of cofactors. Note the transpose: entry (i, j) of A⁻¹ is the cofactor C_ji divided by det A. A permutation matrix has determinant +1 or −1, depending on whether an even or odd number of row swaps reorders it to I. Because det A is the product of the eigenvalues, structured matrices are often easiest that way. For example, the n×n all-ones matrix J has eigenvalues n, 0, …, 0, so J − I has eigenvalues n − 1, −1, …, −1.

For a tridiagonal matrix, expanding along the first row and then expanding the leftover minor gives a recurrence of the form Dₙ = a·Dₙ₋₁ − bc·Dₙ₋₂. Compute D₁ and D₂ directly, then iterate.

Key ideas

• det A ≠ 0 ⇔ A is invertible.
• Cofactor expansion works along any row or column. Pick one with many zeros.
• Row swaps flip the sign; scaling a row scales the determinant; adding a multiple of a row changes nothing.
• Triangular matrices: determinant is the product of the diagonal.
• |det A| is the factor by which A scales areas or volumes.

Formulas and rules

Common mistakes

• Writing det(A + B) = det A + det B.
• Forgetting det(cA) uses cⁿ, not c.

Words to know

Singular matrix

det A = 0; not invertible.

det of a triangular matrix

The product of the diagonal entries.

Effect of swapping two rows on det

It changes sign.

det(AB)

det A · det B

Determinant of a scalar multiple

det(cA) = cⁿ·det A for an n×n matrix.

Geometric meaning of |det A|

How much A scales area (2D) or volume (3D).

Determinant (2×2)

ad − bc

Cofactor expansion

Multiply entries of a row or column by their signed cofactors and add.

Cofactor

(−1)ⁱ⁺ʲ times the determinant left after deleting row i and column j.

Row scaling and det

Multiplying a row by k multiplies the determinant by k.

Row replacement and det

Adding a multiple of one row to another leaves det unchanged.

det(Aᵀ)

det A

Cramer's rule

xᵢ = det Aᵢ / det A

Sample practice questions

29 more questions like these, from level 1 to 5, are in the practice set.