5.1/5.2 - Determinants

Let T : ℝⁿ → ℝⁿ be a linear transform and A be the square matrix defined so that T(x) = Ax. Then the absolute value of the determinant of A, denoted det(A), measures the change in volume under T. This means that if S is a shape of volume V then T(S) = {T(s) : s ∈ ℝⁿ} has volume |det (A)|V.

Definition: For any n ∈ ℕ, The determinant is the unique function from the set of square matrices of size n so the real numbers with the following 3 properties:

• det (I_n) = 1,
• When viewing a square matrices as a list of n column vectors, the determinant is n-linear. This means det ([a₁, …, ca_i + db_i, …, a_n]) = cdet ([a₁, …, a_i, …, a_n]) + ddet ([a₁, …, b_i, …, a_n]).
• If this there is a column of zero, then the determinant is zero.

The determinant of a linear transform T : ℝⁿ → ℝⁿ is the determinant of its associated matrix.

Properties:

• det (AB) = det (A)det (B),
• det (A^t) = det (A),
• If A is upper (or lower) triangular, then det (A) is the product of the diagonal,
• det (cA) = cⁿdet (A),
• If A has positive nullity, then det (A) = 0.

Computation: There is a thing called cofactor expansion. It is terrible but we will learn it. In practice, another method is used like LU-decomposition. There LU decompositon of a matrix A is A = LU where L is lower trianguluar and U is upper triangular. Then det (A) = det (L)det (U), where det (L) and det (U) is the product of the diagonal. Give n = 2 and n = 3 shortcuts and the general cofactor formula. Note that you can expand along any row or column.

Notation: Often you denote the determinant of a matrix by replacing the square brackets by straight lines.

Examples: Compute the determinant in the following cases:

• T(x, y) = (x + y, y)
• T(x, y) = (3x + y,  − y). Also what is the determinant of T^− 1 here? (demonstrate that det (A^− 1) = det (A)^− 1 here)
• Here is a 3 × 3 example
• Here is a 4 × 4 example that I won’t actually work out fully

Theorem: Let S = {a₁, …, a_n} be a set of vectors in ℝⁿ, let A = [a₁ … a_n] and T : ℝⁿ → ℝⁿ be given by T(x) = Ax. Then the following are equivalent:

• S spans ℝⁿ,
• S is linearly independent,
• S is a basis for ℝⁿ,
• Ax = b has a unique solution for every b ∈ ℝⁿ,
• T is onto,
• T is one-to-one,
• ker T = {0},
• range(T) = ℝⁿ,
• col(A) = ℝⁿ,
• row(A) = ℝⁿ,
• rank(A) = n,
• nullity(A) = 0,
• Any echelon form of A has no zero entries on the diagonal,
• The reduced echelon form of A is the identity matrix,
• det (A) ≠ 0,

tldr: Determinant tells you about change of volume. It is a test of invertibility.