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Analysis & Linear Algebra

Determinants and Matrix Tricks

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1. Determinants, trace, block matrices, rank inequalities

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Determinant

NPTEL · Matrix Theory

Properties of determinants

NPTEL · Matrix Theory

Multiplicativity and the block formulas follow from these.

Determinants, trace and block matrices free

In plain words

The determinant measures how much a map scales volume, and whether it flips orientation. Zero means space gets flattened onto something thinner, so nothing can be undone afterwards.

Core identities

  • det(AB) = det A det B; det(Aᵀ) = det A; det(A⁻1)=1/det⁡A;det⁡(^{1}) = 1/\det A; \det(cA)=cndet⁡A) = c^{n} \det A. −det⁡A=∏λi- \det A = \prod\lambda_{i}, trace A=∑λi(A = \sum\lambda_{i} (over the algebraic closure, with multiplicity).
  • trace(AB) = trace(BA) — hence trace is similarity invariant, and AB − BA never equals I (trace 0 vs n).
  • det(I + AB) = det(I + BA), valid even when A is m×n and B is n×m.
  • Characteristic polynomials of AB and BA agree for square A, B.

Block matrices

Block matrix Determinant
[[A, B],[0, D]] (block triangular) det A · det D
[[A, B],[C, D]] with A invertible det A · det(D − CA⁻1B)(^{1}B) (Schur complement)
[[A, B],[C, D]] with CD = DC det(AD − BC)
[[0, I],[I, 0]] (n blocks) (−1)n(-1)^{n}

Rank-one updates

  • det(I + uvᵀ) = 1 + vᵀu (matrix determinant lemma).
  • Eigenvalues of uvᵀ: vᵀu once and 0 with multiplicity n − 1; trace = vᵀu.
  • Sherman–Morrison: (A + uvᵀ)⁻1=A^{1} = A⁻1−A^{1} - A⁻1^{1}uvᵀA⁻1/(1+vTA^{1}/(1 + v^{T}A⁻1u)^{1}u).

Special determinants

Matrix Determinant
Vandermonde (xij−1)(x_{i}^{j-1}) ∏i<j(xj−xi)\prod_{i<j}(x_{j} - x_{i})
Circulant ∏f(ωk)\prod f(\omega^{k}) with f the first-row polynomial
aI + bJ (J all-ones, n×n) an−1(aa^{n-1}(a + nb)
tridiagonal three-term recursion

Rank inequalities

rank(A + B) ≤ rank A + rank B; rank(AB) ≤ min; Sylvester: rank(AB) ≥ rank A + rank B − n.

Key takeaways

  • det(I + AB) = det(I + BA) collapses many rectangular problems.
  • Rank-one matrices: one non-zero eigenvalue vᵀu, everything else 0.
  • trace(AB) = trace(BA) proves AB − BA ≠ I instantly.

Block matrices: which determinant formulas are actually trueinteractive

This visual is part of Trap Analysis

The triangular block rule, and the tempting formula that fails unless the blocks commute.

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Exam focus: Block formulas and the trace/determinant identities turn hard computations into one-liners. det(I + AB) = det(I + BA) is the single most useful trick here.

Practice

Part B2024 Decembershow ▾
Let V be the R−\mathbb{R}-vector space of 5 × 5 real matrices. Let S = {AB − BA | A, B ∈ V} and W denote the subspace of V spanned by S. Let T:V→RT : V \to \mathbb{R} be the linear transformation mapping a matrix A to its trace. Which of the following statements is true?
  1. A.W = ker(T)✓ Correct
  2. B.W ⊊ ker(T)
  3. C.W ∩ ker(T) ⊊ W
  4. D.W ∩ ker(T) ⊊ ker(T)

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let M2(R)M_{2}(\mathbb{R}) denote the R−\mathbb{R}-vector space of 2 × 2 matrices with real entries. Let A = [[1, 2], [0, 3]] and B = [[−1, 0], [1, 5]]. Define a linear transformation T:M2(R)→M2(R)T : M_{2}(\mathbb{R}) \to M_{2}(\mathbb{R}) by T(X) = AXBᵗ, where Bᵗ denotes the transpose of the matrix B. Which of the following statements are true?
  1. A.det(T) = 225✓ Correct
  2. B.det(T) = −225
  3. C.Trace(T) = 16✓ Correct
  4. D.Trace(T) = −16

The worked solution and the per-option analysis are in Trap Analysis.

Part C2024 Decembershow ▾
Let A, B be 2 × 2 matrices with real entries, and M = AB − BA. Let I2I_{2} denote the 2 × 2 identity matrix. Which of the following statements are necessarily true?
  1. A.If A and B are upper triangular, then M is diagonalizable over R\mathbb{R}.
  2. B.If A and B are diagonalizable over R\mathbb{R}, then M is diagonalizable over R\mathbb{R}.
  3. C.If A and B are diagonalizable over R\mathbb{R}, then there exists λ∈R\lambda \in \mathbb{R} such that M=λI2M = \lambda{}I_{2}.
  4. D.There exists λ∈R\lambda \in \mathbb{R} such that M2=λI2M^{2} = \lambda{}I_{2}.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let A be m×n and B be n×m real matrices. Which of the following are true?
  1. A.trace(AB) = trace(BA)✓ Correct
  2. B.det(I_m + AB) = det(I_n + BA)✓ Correct
  3. C.AB and BA have the same non-zero eigenvalues✓ Correct
  4. D.rank(AB) = rank(BA)

The worked solution and the per-option analysis are in Trap Analysis.

Part Cshow ▾
Let A and B be n × n real matrices. Which of the following rank statements are always true?
  1. A.rank(AB) ≤ min(rank A, rank B).✓ Correct
  2. B.rank(A + B) ≤ rank A + rank B.✓ Correct
  3. C.rank(AB) = rank(BA).
  4. D.If A is invertible, then rank(AB) = rank B.✓ Correct

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
Let A be a 3 × 3 real matrix with det A = 2. Then det(adj A) equals
  1. A.2
  2. B.4✓ Correct
  3. C.8
  4. D.1/2

The worked solution and the per-option analysis are in Trap Analysis.

Part Bshow ▾
Let A = J − I where J is the 5×5 all-ones matrix. Then det A equals
  1. A.4✓ Correct
  2. B.−4
  3. C.1
  4. D.0

Solution

A = −I + J has eigenvalues −1 + 5 = 4 (once) and −1 (four times), so det⁡=4⋅(−1)4=4\det = 4\cdot(-1)^{4} = 4.

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