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Module 2/Determinants

Leibniz Definition, Block Matrices & Cramer's Rule

An enrichment tour beyond the three axioms (D1)-(D3): the Leibniz (permutation) formula for the determinant, why block-triangular matrices simply multiply their determinants, and how Cramer's rule and the Vandermonde determinant turn determinants into problem-solving 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.

Compute the determinant of the block-triangular matrix M=[3172245900210035]M=\begin{bmatrix}3&1&7&2\\2&4&5&9\\0&0&2&1\\0&0&3&5\end{bmatrix} using Theorem 6.4.

Evaluate the 3×33\times3 Vandermonde determinant with nodes λ0=1, λ1=2, λ2=5\lambda_0=1,\ \lambda_1=2,\ \lambda_2=5: V=[1111251425]V=\begin{bmatrix}1&1&1\\1&2&5\\1&4&25\end{bmatrix}.

What you’ll be able to do

  • State the Leibniz (permutation) definition of the determinant (Definitions 6.4-6.5) and compute sgn⁡P=(−1)(# inversions)\operatorname{sgn}P=(-1)^{(\#\,\text{inversions})} for a given pattern.
  • Use the Leibniz formula det⁡(A)=∑P(sgn⁡P)(prod⁡P)\det(A)=\sum_P(\operatorname{sgn}P)(\operatorname{prod}P) to evaluate small determinants and to explain why det⁡(A⊤)=det⁡(A)\det(A^\top)=\det(A).
  • Apply Theorem 6.4 to evaluate block-triangular determinants, and recognize (Remark 6.2) that det⁡[ABCD]=det⁡A det⁡D−det⁡B det⁡C\det\begin{bmatrix}A&B\\C&D\end{bmatrix}=\det A\,\det D-\det B\,\det C fails in general.
  • Solve 2×22\times2 and 3×33\times3 linear systems with Cramer's rule, xi=det⁡(Ai)/det⁡(A)x_i=\det(A_i)/\det(A).
  • Evaluate a Vandermonde determinant via ∏0≤i<j≤n(λj−λi)\prod_{0\le i<j\le n}(\lambda_j-\lambda_i) and state when it is nonzero.

In your course

· MATH2015 · Linear Algebra & Probability
§6.1 The determinant function§6.2 Properties of determinant
  • Definition 6.4Patterns, inversions, signature
    A pattern picks one entry per row and column; sgn⁡P=(−1)(# inversions)\operatorname{sgn}P=(-1)^{(\#\,\text{inversions})}.
  • Definition 6.5Leibniz formula
    det⁡(A)=∑P(sgn⁡P)(prod⁡P)\det(A)=\sum_P(\operatorname{sgn}P)(\operatorname{prod}P) over all n!n! patterns.
  • Theorem 6.3Uniqueness of the determinant
    Exactly one det⁡:Rn×n→R\det:\mathbb{R}^{n\times n}\to\mathbb{R} satisfies (D1)-(D3).
  • Theorem 6.4Determinant of a block matrix
    det⁡[AB0C]=det⁡(A)det⁡(C)\det\begin{bmatrix}A&B\\0&C\end{bmatrix}=\det(A)\det(C) for square A,CA,C.
  • Remark 6.2Naive block formula fails
    det⁡[ABCD]=det⁡(A)det⁡(D)−det⁡(B)det⁡(C)\det\begin{bmatrix}A&B\\C&D\end{bmatrix}=\det(A)\det(D)-\det(B)\det(C) does not always hold.
  • Section 6.2.1, Ex. 3Vandermonde determinant
    det⁡(V)=∏0≤i<j≤n(λj−λi)\det(V)=\prod_{0\le i<j\le n}(\lambda_j-\lambda_i).
  • Section 6.2.1, Ex. 4Cramer's Rule
    xi=det⁡(Ai)/det⁡(A)x_i=\det(A_i)/\det(A), with AiA_i = AA with column ii replaced by bb.
Enrichment: Leibniz/permutation definition, block matrices, Cramer's rule
1

Patterns, inversions, and the signature (Definition 6.4)

Beyond the axioms (D1)-(D3), the determinant can be built combinatorially. A pattern in an n×nn\times n matrix is a choice of nn entries with exactly one in each row and exactly one in each column, so there are n!n! patterns. Writing the chosen column of row ii as σ(i)\sigma(i), a pattern is just a permutation σ\sigma. Two chosen entries are inverted when one lies to the right and above the other, i.e. a pair of rows i<ji<j with σ(i)>σ(j)\sigma(i)>\sigma(j). The signature is sgn⁡P=(−1)(# inversions in P)\operatorname{sgn}P=(-1)^{(\#\,\text{inversions in }P)}, so interchanging the columns of two chosen entries flips the sign.

2

The Leibniz formula (Definition 6.5)

The determinant is det⁡(A)=∑P(sgn⁡P)(prod⁡P)\det(A)=\sum_{P}(\operatorname{sgn}P)(\operatorname{prod}P), the signed sum over all n!n! patterns, where prod⁡P\operatorname{prod}P is the product of the entries of PP. Patterns with an even number of inversions are added and those with an odd number are subtracted. For n=2n=2 the two patterns give adad (no inversion, ++) and bcbc (one inversion, −-), recovering ad−bcad-bc; for n=3n=3 it reproduces the Rule of Sarrus. The formula also makes structural facts transparent: transposing swaps 'right-and-above' for 'left-and-below' without changing the inversion count, so det⁡(A⊤)=det⁡(A)\det(A^\top)=\det(A) (Theorem 6.5).

3

Block matrices (Theorem 6.4 & Remark 6.2)

If the lower-left block is zero and A,CA,C are square (any sizes, any BB), then det⁡[AB0C]=det⁡(A)det⁡(C)\det\begin{bmatrix}A&B\\0&C\end{bmatrix}=\det(A)\det(C). Leibniz proof: any pattern using an entry of the zero block contributes 00, so every surviving pattern splits into a pattern of AA and a pattern of CC -- products multiply and inversion counts add, hence signatures multiply too. Warning (Remark 6.2): the tempting 'treat the blocks like scalars' rule det⁡[ABCD]=det⁡(A)det⁡(D)−det⁡(B)det⁡(C)\det\begin{bmatrix}A&B\\C&D\end{bmatrix}=\det(A)\det(D)-\det(B)\det(C) is false in general once C≠0C\neq0.

4

Cramer's rule and the Vandermonde determinant

When det⁡(A)≠0\det(A)\neq0, the system Ax=bAx=b has the unique solution xi=det⁡(Ai)det⁡(A)x_i=\dfrac{\det(A_i)}{\det(A)}, where AiA_i is AA with its ii-th column replaced by bb. It is elegant but, as the notes caution, impractical for large systems. A determinant worth knowing in closed form is the Vandermonde determinant: for nodes λ0,…,λn\lambda_0,\dots,\lambda_n it equals ∏0≤i<j≤n(λj−λi)\prod_{0\le i<j\le n}(\lambda_j-\lambda_i), which is nonzero exactly when the nodes are distinct.

Theorem 6.3 - Uniqueness of the determinant

There is exactly one function det⁡:Rn×n→R\det:\mathbb{R}^{n\times n}\to\mathbb{R} satisfying (D1) det⁡(In)=1\det(I_n)=1, (D2) linearly dependent rows give 00, and (D3) linearity in each row.

Intuition. Row reduction turns any matrix into an upper-triangular one using operations whose effect on the determinant is fixed by (D1)-(D3) (Theorem 6.1: scaling multiplies, row-addition preserves, interchange flips the sign), so the value is forced to be ±\pm the product of the pivots. Existence comes from the Leibniz/cofactor formula (Lemma 6.1), which is shown to satisfy the three axioms.
Theorem 6.4 - Determinant of a block matrix

If AA and CC are square (not necessarily the same size), then det⁡[AB0C]=det⁡(A)det⁡(C)\det\begin{bmatrix}A&B\\0&C\end{bmatrix}=\det(A)\det(C) for every BB.

Intuition. In the Leibniz sum, any pattern touching the zero block dies; the survivors factor as (a pattern of AA) and (a pattern of CC), with products multiplying and inversions adding. Caution: with a nonzero lower-left block the naive det⁡(A)det⁡(D)−det⁡(B)det⁡(C)\det(A)\det(D)-\det(B)\det(C) need not equal the determinant (Remark 6.2).
Cramer's Rule (Section 6.2.1, Ex. 4)

If AA is n×nn\times n with det⁡(A)≠0\det(A)\neq0, the unique solution of Ax=bAx=b is xi=det⁡(Ai)det⁡(A)x_i=\dfrac{\det(A_i)}{\det(A)}, where AiA_i is AA with its ii-th column replaced by bb.

Intuition. Write b=Ax=∑kxkckb=Ax=\sum_k x_k c_k in column ii of AiA_i. By column-linearity (Proposition 6.3) every term except xicix_i c_i produces a repeated column, hence determinant 00, leaving det⁡(Ai)=xidet⁡(A)\det(A_i)=x_i\det(A). For n=2n=2 this is x=1Δdet⁡[pbqd]x=\tfrac{1}{\Delta}\det\begin{bmatrix}p&b\\q&d\end{bmatrix}, y=1Δdet⁡[apcq]y=\tfrac{1}{\Delta}\det\begin{bmatrix}a&p\\c&q\end{bmatrix}. Elegant, but impractical for large nn.
Vandermonde determinant (Section 6.2.1, Ex. 3)

For the Vandermonde matrix VV with nodes λ0,…,λn\lambda_0,\dots,\lambda_n (each column kk being 1,λk,λk2,…,λkn1,\lambda_k,\lambda_k^2,\dots,\lambda_k^n), det⁡(V)=∏0≤i<j≤n(λj−λi)\det(V)=\prod_{0\le i<j\le n}(\lambda_j-\lambda_i).

Intuition. As a polynomial in the nodes it vanishes whenever two nodes coincide (two equal columns force det⁡=0\det=0), so each factor (λj−λi)(\lambda_j-\lambda_i) divides it; matching degree and leading term pins the constant to 11. Consequently det⁡(V)≠0\det(V)\neq0 iff the nodes are distinct.

Worked examples

Example 1

Use the Leibniz definition to compute det⁡(A)\det(A) for A=[201130412]A=\begin{bmatrix}2&0&1\\1&3&0\\4&1&2\end{bmatrix} by listing all 3!=63!=6 patterns.

  1. 1

    Each pattern picks one entry per row and column, so it is given by the columns (σ(1),σ(2),σ(3))(\sigma(1),\sigma(2),\sigma(3)) chosen in rows 1,2,31,2,3. An inversion is a pair where a larger column precedes a smaller one.

  2. 2

    σ=(1,2,3)\sigma=(1,2,3): entries a11a22a33=2⋅3⋅2=12a_{11}a_{22}a_{33}=2\cdot3\cdot2=12; 00 inversions, sgn⁡=+1\operatorname{sgn}=+1; contribution +12+12.

  3. 3

    σ=(1,3,2)\sigma=(1,3,2): a11a23a32=2⋅0⋅1=0a_{11}a_{23}a_{32}=2\cdot0\cdot1=0.

  4. 4

    σ=(2,1,3)\sigma=(2,1,3): a12a21a33=0⋅1⋅2=0a_{12}a_{21}a_{33}=0\cdot1\cdot2=0.

  5. 5

    σ=(2,3,1)\sigma=(2,3,1): a12a23a31=0⋅0⋅4=0a_{12}a_{23}a_{31}=0\cdot0\cdot4=0.

  6. 6

    σ=(3,1,2)\sigma=(3,1,2): a13a21a32=1⋅1⋅1=1a_{13}a_{21}a_{32}=1\cdot1\cdot1=1; inversions '33 before 11' and '33 before 22' give 22, so sgn⁡=+1\operatorname{sgn}=+1; contribution +1+1.

  7. 7

    σ=(3,2,1)\sigma=(3,2,1): a13a22a31=1⋅3⋅4=12a_{13}a_{22}a_{31}=1\cdot3\cdot4=12; inversions '33 before 22', '33 before 11', '22 before 11' give 33, so sgn⁡=−1\operatorname{sgn}=-1; contribution −12-12.

  8. 8

    Add all patterns: det⁡(A)=12+0+0+0+1−12=1\det(A)=12+0+0+0+1-12=1.

Answer. det⁡(A)=1\det(A)=1.
Example 2

Compute det⁡(M)\det(M) for the block matrix M=[2112133400310024]M=\begin{bmatrix}2&1&1&2\\1&3&3&4\\0&0&3&1\\0&0&2&4\end{bmatrix}.

Example 3

Solve {3x+2y=7x+4y=9\begin{cases}3x+2y=7\\x+4y=9\end{cases} using Cramer's rule.