Determinants

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Determinants — Quick Facts Determinant of 2×2: |a b; c d| = ad - bc Minor Mᵢⱼ: determinant obtained by deleting ith row and jth column; Cofactor Cᵢⱼ = (-1)^(i+j) Mᵢⱼ Laplace expansion: det A = Σ Cᵢⱼ aᵢⱼ = Σ Cᵢⱼ aᵢⱼ (along any row/column) Product: det(AB) = det(A)·det(B); det(A⁻¹) = 1/det(A); det(Aᵀ) = det(A) ⚡ Exam tip: Use row/column operations to simplify before expanding — zero rows/columns are gold

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Determinants — Study Guide

Definition and Basic Properties

A determinant is a scalar value computed from a square matrix. For 2×2: $$\begin{vmatrix} a & b \ c & d \end{vmatrix} = ad - bc$$

For 3×3, use Sarrus rule or cofactor expansion: $$\begin{vmatrix} a & b & c \ d & e & f \ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$$

Key Properties:

1. det(Aᵀ) = det(A) — rows and columns interchangeable
2. Swapping two rows/columns changes sign of determinant
3. Multiplying a row by scalar k multiplies det by k
4. Adding multiple of one row to another does NOT change det
5. If two rows are identical or proportional, det = 0
6. det(I) = 1, det(0) = 0

Cofactor and Adjoint

Minor Mᵢⱼ: determinant after deleting i-th row, j-th column Cofactor Cᵢⱼ = (-1)^(i+j) Mᵢⱼ

Adjoint of A: adj(A) is matrix of cofactors transposed $$A \cdot adj(A) = adj(A) \cdot A = det(A) \cdot I$$

This gives formula for inverse: $$A^{-1} = \frac{adj(A)}{det(A)} \quad \text{provided } det(A) \neq 0$$

System of Linear Equations (Cramer’s Rule)

For Ax = b where A is n×n with det(A) ≠ 0: $$x_i = \frac{det(A_i)}{det(A)}$$

where Aᵢ is matrix A with i-th column replaced by b.

For 2 equations: $$x = \frac{\begin{vmatrix} c_1 & b_1 \ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix}}, \quad y = \frac{\begin{vmatrix} a_1 & c_1 \ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \ a_2 & b_2 \end{vmatrix}}$$

Cramer’s rule is theoretically elegant but computationally inefficient for large systems.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Determinants — Comprehensive Notes

Matrix Representation and Determinant via Permutations

For n×n matrix A = [aᵢⱼ], determinant can be written as: $$det(A) = \sum_{\sigma \in S_n} \text{sgn}(\sigma) \prod_{i=1}^{n} a_{i,\sigma(i)}$$

where Sₙ is set of all permutations of {1,2,…,n} and sgn(σ) is parity (+1 for even, -1 for odd permutation). This definition is rarely used computationally but essential for proving determinant properties.

For 3×3, the 6 permutations give the expansion formula shown earlier. For 4×4, there are 24 terms — impractical to expand directly, hence the cofactor method is preferred.

Elementary Row/Column Operations and Rank

Elementary matrices (from row operations):

• Type I: swap rows i and j → det changes sign
• Type II: multiply row i by scalar k → det multiplied by k
• Type III: add k× row j to row i → det unchanged

Using operations to simplify matrix to triangular form makes determinant trivial (product of diagonal entries).

** Echelon form**: Using only Type III operations (which preserve det), any non-singular matrix can be reduced to upper triangular. det = product of diagonal after reduction.

Rank via determinants: Rank of A = size of largest non-zero minor. If all p×p minors are zero but some (p-1)×(p-1) minor is non-zero, rank = p-1.

Vandermonde Determinant

$$V = \begin{vmatrix} 1 & 1 & 1 & … & 1 \ x_1 & x_2 & x_3 & … & x_n \ x_1^2 & x_2^2 & x_3^2 & … & x_n^2 \ \vdots & \vdots & \vdots & & \vdots \ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & … & x_n^{n-1} \end{vmatrix} = \prod_{1 \leq i < j \leq n} (x_j - x_i)$$

This product form is zero when any xᵢ = xⱼ (two equal x-values → rows become identical). This determinant appears in interpolation problems.

Circulant Determinants

A circulant matrix has each row a cyclic shift of the previous: $$C = \begin{vmatrix} a_0 & a_1 & a_2 & … & a_{n-1} \ a_{n-1} & a_0 & a_1 & … & a_{n-2} \ \vdots & & & & \vdots \ a_1 & a_2 & a_3 & … & a_0 \end{vmatrix}$$

det(C) = ∏ rₖ where rₖ = a₀ + a₁ωₖ + a₂ωₖ² + … + aₙ₋₁ωₖⁿ⁻¹ and ωₖ = e^(2πik/n) are nth roots of unity.

For n = 3: eigenvalues are evaluated at cube roots of unity.

Laplace Expansion (Cofactor Expansion)

Expanding along i-th row: $$det(A) = \sum_{j=1}^{n} (-1)^{i+j} a_{ij} M_{ij}$$

Shortcut: Multiply row by cofactor of another row and sum = 0: $$\sum_{j} a_{ij} C_{kj} = 0 \text{ for } i \neq k$$

This is because adding a multiple of one row to another doesn’t change determinant (but the expansion gives the determinant of a matrix with row k replaced by row i, which has two identical rows → determinant = 0).

Similarly: $\sum_{i} a_{ij} C_{ik} = 0$ for j ≠ k.

This property is useful for evaluating determinants by strategic zero-creation.

Block Matrices

For block matrix $\begin{pmatrix} A & B \ C & D \end{pmatrix}$ where A and D are square:

• If A is invertible: det = det(A)·det(D - CA⁻¹B)
• If D is invertible: det = det(D)·det(A - BD⁻¹C)
• If AC = CA (commute): det = det(AD - BC)

Special case (upper triangular blocks): det = det(A)·det(D)

Jacobians as Determinants

When transforming variables from (x,y) to (u,v) via x = f(u,v), y = g(u,v): $$J = \frac{\partial(x,y)}{\partial(u,v)} = \begin{vmatrix} \partial x/\partial u & \partial x/\partial v \ \partial y/\partial u & \partial y/\partial v \end{vmatrix}$$

Jacobian determinant appears in change of variables for double integrals: $$\iint_R f(x,y) dx dy = \iint_S f(x(u,v), y(u,v)) |J| du dv$$

JEE Application: Coordinate transformations (Cartesian ↔ polar, spherical, etc.)

Differentiation of Determinants

If each element of A(x) is differentiable w.r.t. x: $$\frac{d}{dx} det(A) = det(A) \cdot tr(A^{-1} \cdot \frac{dA}{dx})$$

where tr is trace (sum of diagonal elements).

For matrix with functions on diagonal only: d/dx det(diag(f₁, f₂, …, fₙ)) = det(A)·Σ(fᵢ’/fᵢ)

This is useful in differential equations involving determinants.

Product Rule via Determinants

Cauchy-Binet formula: For A (m×n) and B (n×m) with m ≤ n: $$det(AB) = \sum_{1 \leq j_1 < j_2 < … < j_m \leq n} det(A[:, j_1…j_m]) \cdot det(B[j_1…j_m, :])$$

When m = n, this reduces to det(AB) = det(A)·det(B). When m < n, det(AB) is sum over all m×m minors.

⚡ Exam tips for JEE Advanced:

1. Create zeros before expanding — a row of all zeros means determinant = 0
2. For symmetric problems, try to show det > 0 or < 0 based on eigenvalues
3. In eigenvalue problems, det(A - λI) = 0 gives characteristic equation
4. Common pattern: det of matrix with variables can often be factored — try substitution of simple values (x=0, x=1, etc.) to find factors
5. For 3×3 with variables, determinant is at most degree 3 in any variable
6. Adjoint property: A·adj(A) = det(A)I → if det(A)=1, then A⁻¹ = adj(A)
7. In geometry problems, triangle area = (1/2)|det(v₁-v₀, v₂-v₀)| where vertices are vectors from origin
8. When determinant appears in limits/integrals, try to identify it as product of differences (Vandermonde pattern)
9. For functional determinants in change of variables, J appears as |J| — don’t forget absolute value

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