Determinants and Matrix Tricks

Determinant

NPTEL · Matrix Theory

Properties of determinants

NPTEL · Matrix Theory

Multiplicativity and the block formulas follow from these.

Block matrices: which determinant formulas are actually trueinteractive

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

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Determinants, trace and block matrices

Core identities

det(AB) = det A det B; det(Aᵀ) = det A; det(A⁻ $^{1}) = 1/ det A; det ($ cA $) = c^{n} det A$ . $- det A = \prod λ_{i}$ , trace $A = \sum λ_{i} ($ 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

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Part Bshow ▾

Let A = J − I where J is the 5×5 all-ones matrix. Then det A equals

A.4✓
B.−4
C.1
D.0

Solution

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

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Part Cshow ▾

Let A be m×n and B be n×m real matrices. Which of the following are true?

A.trace(AB) = trace(BA)✓
B.det(I_m + AB) = det(I_n + BA)✓
C.AB and BA have the same non-zero eigenvalues✓
D.rank(AB) = rank(BA)

Solution

(d) fails: with A a 1×2 row and B a 2×1 column, AB is 1×1 (rank 1 if AB ≠ 0) while BA is 2×2 of rank 1 — but take A = [1 0], B = [0 1]ᵀ: AB = 0 has rank 0, BA has rank 1.

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Part Cshow ▾

Let A and B be n × n real matrices. Which of the following rank statements are always true?

A.rank(AB) ≤ min(rank A, rank B).✓
B.rank(A + B) ≤ rank A + rank B.✓
C.rank(AB) = rank(BA).
D.If A is invertible, then rank(AB) = rank B.✓

Solution

(1) The column space of AB sits inside that of A, and its row space inside that of B. (2) The column space of A + B lies in the sum of the two column spaces. (3) fails: with A having a single 1 in position (1,2) and B a single 1 in position (2,2), AB has rank 1 while BA = 0. (4) Multiplying by an invertible matrix is a change of basis and preserves rank.

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Part Bshow ▾

Let A be a 3 × 3 real matrix with det A = 2. Then det(adj A) equals

A.2
B.4✓
C.8
D.1/2

Solution

A·adj A = (det A)I gives det A · det(adj $A) = (det A)^{3}$ , so det(adj A) = (det $A)^{n - 1}$ $= 2^{2} = 4$ for n = 3.

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1. Determinants, trace, block matrices, rank inequalities

Block matrices: which determinant formulas are actually trueinteractive

Determinants, trace and block matrices

Core identities