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Module 2/Determinants

Properties of Determinants

The determinant is more than a number you grind out by row reduction: it is a multiplicative invariant that detects invertibility and behaves predictably under products, powers, inverses, transposes, and change of basis. This lesson (§6.2) collects the key algebraic properties of det⁡\det and turns them into fast computational tools.

Before you start — give these a try

Attempting first primes your brain for the lesson — even if you miss. Nothing is graded or saved; it's just a warm-up.

Let AA and BB be 3×33\times 3 matrices with det⁡(A)=2\det(A)=2 and det⁡(B)=−4\det(B)=-4. Compute det⁡(A2B)\det(A^2 B).

Let AA be a 3×33\times 3 matrix with det⁡(A)=6\det(A)=6. Compute det⁡(2A)\det(2A).

What you’ll be able to do

  • Apply Theorem 6.2 to decide invertibility from the determinant, linking det⁡(A)=0\det(A)=0 with rank deficiency, a nontrivial kernel, and linearly dependent rows or columns.
  • Use the multiplicative property det⁡(AB)=det⁡(A)det⁡(B)\det(AB)=\det(A)\det(B) to compute determinants of products, powers AkA^k, and inverses A−1A^{-1}.
  • Use det⁡(AT)=det⁡(A)\det(A^T)=\det(A) together with Proposition 6.3 to move freely between row and column arguments.
  • Compute det⁡(λA)=λndet⁡(A)\det(\lambda A)=\lambda^n\det(A) and explain why the determinant is not additive, i.e. det⁡(A+B)≠det⁡(A)+det⁡(B)\det(A+B)\neq\det(A)+\det(B) in general.
  • Recognize that similar matrices share the same determinant (Corollary 6.1).

In your course

· MATH2015 · Linear Algebra & Probability
§6.2 Properties of determinant
  • Theorem 6.2Determinant, rank, and invertibility
    For A∈Rn×nA\in\mathbb{R}^{n\times n}: det⁡(A)=0  ⟺  rank⁡(A)<n  ⟺  dim⁡ker⁡(A)>0  ⟺  \det(A)=0 \iff \operatorname{rank}(A)<n \iff \dim\ker(A)>0 \iff the columns (rows) are linearly dependent. Equivalently det⁡(A)≠0  ⟺  rank⁡(A)=n  ⟺  ker⁡(A)={0}  ⟺  A\det(A)\neq 0 \iff \operatorname{rank}(A)=n \iff \ker(A)=\{0\} \iff A is invertible.
  • Theorem 6.5Determinant of the transpose
    If AA is square, then det⁡(AT)=det⁡(A)\det(A^T)=\det(A).
  • Theorem 6.6Determinants of products and powers
    For n×nn\times n matrices A,BA,B and a positive integer kk: det⁡(AB)=det⁡(A)det⁡(B)\det(AB)=\det(A)\det(B) and det⁡(Ak)=[det⁡(A)]k\det(A^k)=[\det(A)]^k.
  • Theorem 6.7Determinant of an inverse
    If AA is invertible, then det⁡(A−1)=1/det⁡(A)\det(A^{-1})=1/\det(A).
  • Corollary 6.1Determinant of similar matrices
    If AA is similar to BB (i.e. A=SBS−1A=SBS^{-1} for some invertible SS), then det⁡(A)=det⁡(B)\det(A)=\det(B).
  • Proposition 6.3Column versions of the row properties
    All row properties hold for columns: dependent columns give det⁡=0\det=0; scaling a column by λ\lambda scales det⁡\det by λ\lambda; adding λ\lambda times one column to another leaves det⁡\det unchanged; swapping two columns reverses the sign.
These properties rest on the three defining properties (D1)-(D3) of Definition 6.1; Theorems 6.5 and 6.6 are proved with the Leibniz (pattern) definition (Definition 6.5), while the elementary-row-operation rules (Theorem 6.1) and the triangular-matrix rule (Proposition 6.1) from §6.1 are the computational tools behind them. The identity det⁡(λA)=λndet⁡(A)\det(\lambda A)=\lambda^n\det(A) comes from scaling all nn rows; its special case det⁡(−A)=(−1)ndet⁡(A)\det(-A)=(-1)^n\det(A) is Exercise 1 of §6.2.1.
1

The determinant as an invertibility detector

The single most useful fact about determinants is Theorem 6.2: for A∈Rn×nA\in\mathbb{R}^{n\times n}, det⁡(A)≠0  ⟺  rank⁡(A)=n  ⟺  ker⁡(A)={0}  ⟺  A is invertible,\det(A)\neq 0 \iff \operatorname{rank}(A)=n \iff \ker(A)=\{0\} \iff A\text{ is invertible}, while the opposite case collects all the ways a matrix can fail: det⁡(A)=0  ⟺  rank⁡(A)<n  ⟺  dim⁡ker⁡(A)>0  ⟺  rows (columns) linearly dependent.\det(A)=0 \iff \operatorname{rank}(A)<n \iff \dim\ker(A)>0 \iff \text{rows (columns) linearly dependent}. This is why spotting a zero row, a repeated row, or one row that is a scalar multiple of another lets you declare det⁡(A)=0\det(A)=0 on sight, with no computation.

2

The determinant is multiplicative

det⁡(AB)=det⁡(A)det⁡(B)\det(AB)=\det(A)\det(B) (Theorem 6.6) is the engine behind almost everything else. Powers follow immediately: det⁡(Ak)=[det⁡(A)]k\det(A^k)=[\det(A)]^k. Applying det⁡\det to AA−1=InAA^{-1}=I_n gives det⁡(A)det⁡(A−1)=1\det(A)\det(A^{-1})=1, hence det⁡(A−1)=1/det⁡(A)\det(A^{-1})=1/\det(A) (Theorem 6.7). And similar matrices have equal determinant (Corollary 6.1): det⁡(SBS−1)=det⁡(S)det⁡(B)det⁡(S)−1=det⁡(B)\det(SBS^{-1})=\det(S)\det(B)\det(S)^{-1}=\det(B). Because real numbers commute, you also get det⁡(AB)=det⁡(A)det⁡(B)=det⁡(B)det⁡(A)=det⁡(BA)\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA) even though AB≠BAAB\neq BA in general.

3

Rows and columns play symmetric roles

Theorem 6.5 says det⁡(AT)=det⁡(A)\det(A^T)=\det(A). Since transposing swaps rows and columns without changing the value, every property stated for rows holds verbatim for columns (Proposition 6.3): dependent columns force det⁡=0\det=0; scaling one column by λ\lambda scales det⁡\det by λ\lambda; adding λ\lambda times one column to another leaves det⁡\det unchanged; swapping two columns flips the sign. Scaling all nn rows (or columns) at once gives det⁡(λA)=λndet⁡(A)\det(\lambda A)=\lambda^n\det(A), and in particular det⁡(−A)=(−1)ndet⁡(A)\det(-A)=(-1)^n\det(A).

4

What the determinant is NOT

The determinant is multiplicative, not additive: det⁡(A+B)≠det⁡(A)+det⁡(B)\det(A+B)\neq\det(A)+\det(B) in general. For example A=[1000]A=\begin{bmatrix}1&0\\0&0\end{bmatrix} and B=[0001]B=\begin{bmatrix}0&0\\0&1\end{bmatrix} have det⁡(A)=det⁡(B)=0\det(A)=\det(B)=0, yet A+B=I2A+B=I_2 with det⁡(A+B)=1\det(A+B)=1. Likewise det⁡(λA)=λndet⁡(A)\det(\lambda A)=\lambda^n\det(A), not λdet⁡(A)\lambda\det(A): pulling a scalar out of a matrix raises it to the power nn, once for each row.

Theorem 6.2 - Determinant, rank, and invertibility

Let A∈Rn×nA\in\mathbb{R}^{n\times n}. Then det⁡(A)=0  ⟺  rank⁡(A)<n  ⟺  dim⁡ker⁡(A)>0  ⟺  the columns (rows) of A are linearly dependent.\det(A)=0 \iff \operatorname{rank}(A)<n \iff \dim\ker(A)>0 \iff \text{the columns (rows) of }A\text{ are linearly dependent}. Equivalently, det⁡(A)≠0  ⟺  rank⁡(A)=n  ⟺  ker⁡(A)={0}  ⟺  A is invertible.\det(A)\neq 0 \iff \operatorname{rank}(A)=n \iff \ker(A)=\{0\} \iff A\text{ is invertible}.

Intuition. Gaussian elimination reduces det⁡(A)\det(A) to ±(product of pivots)\pm(\text{product of pivots}). A zero determinant means a missing pivot, i.e. some row is a combination of the others - exactly rank deficiency and a nontrivial kernel. So one number answers the question 'is AA invertible?'
Theorem 6.5 - Determinant of the transpose (with Proposition 6.3)

If AA is a square matrix, then det⁡(AT)=det⁡(A).\det(A^T)=\det(A). Consequently (Proposition 6.3) every row property of the determinant holds for columns too: linearly dependent columns force det⁡=0\det=0; scaling one column by λ\lambda scales the determinant by λ\lambda; adding λ\lambda times one column to another leaves it unchanged; interchanging two columns reverses the sign.

Intuition. In the Leibniz (pattern) definition each term uses one entry per row and per column; transposing relabels rows as columns but keeps the same products and signs. Rows and columns are on equal footing, so you may expand or reduce along whichever is more convenient.
Theorem 6.6 - Determinants of products and powers

For n×nn\times n matrices A,BA,B and a positive integer kk, det⁡(AB)=det⁡(A)det⁡(B)anddet⁡(Ak)=[det⁡(A)]k.\det(AB)=\det(A)\det(B) \qquad\text{and}\qquad \det(A^k)=[\det(A)]^k.

Intuition. The determinant is multiplicative: the volume-scaling factor of a composition ABAB is the product of the two individual factors. A neat corollary is det⁡(AB)=det⁡(A)det⁡(B)=det⁡(B)det⁡(A)=det⁡(BA)\det(AB)=\det(A)\det(B)=\det(B)\det(A)=\det(BA), even though usually AB≠BAAB\neq BA.
Theorem 6.7 - Determinant of an inverse (with Corollary 6.1)

If AA is invertible, then det⁡(A−1)=1det⁡(A).\det(A^{-1})=\frac{1}{\det(A)}.

Intuition. Apply det⁡\det to AA−1=InAA^{-1}=I_n: multiplicativity gives det⁡(A)det⁡(A−1)=det⁡(In)=1\det(A)\det(A^{-1})=\det(I_n)=1. The same idea proves Corollary 6.1: similar matrices A=SBS−1A=SBS^{-1} satisfy det⁡(A)=det⁡(S)det⁡(B)det⁡(S)−1=det⁡(B)\det(A)=\det(S)\det(B)\det(S)^{-1}=\det(B), so similarity preserves the determinant.

Worked examples

Example 1

Let AA and BB be 3×33\times 3 matrices with det⁡(A)=3\det(A)=3 and det⁡(B)=−2\det(B)=-2. Compute det⁡(AB)\det(AB), det⁡(AT)\det(A^T), det⁡(A−1)\det(A^{-1}), det⁡(A3)\det(A^3), det⁡(2A)\det(2A), and det⁡(ATB−1)\det(A^T B^{-1}).

  1. 1

    det⁡(AB)=det⁡(A)det⁡(B)=(3)(−2)=−6\det(AB)=\det(A)\det(B)=(3)(-2)=-6 (Theorem 6.6).

  2. 2

    det⁡(AT)=det⁡(A)=3\det(A^T)=\det(A)=3 (Theorem 6.5).

  3. 3

    det⁡(A−1)=1/det⁡(A)=13\det(A^{-1})=1/\det(A)=\tfrac13 (Theorem 6.7; AA is invertible because det⁡(A)≠0\det(A)\neq 0).

  4. 4

    det⁡(A3)=[det⁡(A)]3=33=27\det(A^3)=[\det(A)]^3=3^3=27 (Theorem 6.6).

  5. 5

    det⁡(2A)=23det⁡(A)=8⋅3=24\det(2A)=2^3\det(A)=8\cdot 3=24: the matrix is 3×33\times 3, so all three rows are scaled, giving a factor 232^3.

  6. 6

    det⁡(ATB−1)=det⁡(AT)det⁡(B−1)=det⁡(A)⋅1det⁡(B)=3⋅1−2=−32\det(A^T B^{-1})=\det(A^T)\det(B^{-1})=\det(A)\cdot\frac{1}{\det(B)}=3\cdot\frac{1}{-2}=-\frac{3}{2}.

Answer. det⁡(AB)=−6, det⁡(AT)=3, det⁡(A−1)=13, det⁡(A3)=27, det⁡(2A)=24, det⁡(ATB−1)=−32\det(AB)=-6,\ \det(A^T)=3,\ \det(A^{-1})=\tfrac13,\ \det(A^3)=27,\ \det(2A)=24,\ \det(A^T B^{-1})=-\tfrac32.
Example 2

Decide whether A=[123246101]A=\begin{bmatrix}1&2&3\\2&4&6\\1&0&1\end{bmatrix} is invertible, and state rank⁡(A)\operatorname{rank}(A) and whether ker⁡(A)={0}\ker(A)=\{0\}.

Example 3

Two square matrices are similar if A=SBS−1A=SBS^{-1} for some invertible SS. (a) Show that det⁡(AB)=det⁡(BA)\det(AB)=\det(BA) for any n×nn\times n matrices A,BA,B. (b) Prove Corollary 6.1: similar matrices have the same determinant.