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Rank & Determinant

Short Notes

Rank

The rank of a matrix is the maximum number of linearly independent rows (or columns). Equivalently, it's the number of non-zero rows in the row echelon form.

  • rank(A) ≤ min(m, n) for an m×n matrix
  • rank(A) = rank(Aᵀ) (row rank = column rank)
  • A matrix has full rank if rank = min(m, n)

Determinant

The determinant is a scalar value that can be computed from a square matrix. It encodes important properties about the linear transformation represented by the matrix.

  • |A| = 0 ⟺ A is singular (not invertible)
  • |A| ≠ 0 ⟺ A is non-singular (invertible)
  • Geometrically: |det(A)| = volume scaling factor of the transformation

Key Theories & Formulas

1. Determinant Properties

Property Result
\(\det(AB) = \det(A) \cdot \det(B)\) Multiplicative
\(\det(A^T) = \det(A)\) Transpose invariant
\(\det(A^{-1}) = \frac{1}{\det(A)}\) Inverse
\(\det(kA) = k^n \det(A)\) Scalar multiplication (n×n matrix)
\(\det(A^n) = [\det(A)]^n\) Powers
Row swap → \(-\det(A)\) Sign change
Row × k → \(k \cdot \det(A)\) Scalar row
\(R_i + kR_j\) → \(\det(A)\) Row addition (unchanged)

2. Computing Determinant

For 2×2 Matrix: $\(\det\begin{pmatrix} a & b \\ c & d \end{pmatrix} = ad - bc\)$

For 3×3 Matrix (Sarrus Rule or Cofactor expansion): $\(\det\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} = a(ei-fh) - b(di-fg) + c(dh-eg)\)$

For upper/lower triangular matrices: $\(\det(A) = \prod_{i} a_{ii}\)$ (product of diagonal elements)

3. Rank Properties

  • \(rank(AB) \leq \min(rank(A), rank(B))\)
  • \(rank(A + B) \leq rank(A) + rank(B)\)
  • \(rank(A) + nullity(A) = n\) (Rank-Nullity Theorem)
  • If A is m×n and B is n×p with AB = 0, then \(rank(A) + rank(B) \leq n\)

4. Rank and Determinant Connection

  • A square matrix A is invertible iff rank(A) = n iff det(A) ≠ 0
  • For any k×k submatrix, if its determinant ≠ 0, the rank of A ≥ k
  • Rank = largest order of non-zero minor

5. Adjoint and Inverse

\[A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\]

where adj(A) = transpose of cofactor matrix.

Example Problems

Problem 1: Find the determinant of \(A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}\)

Solution: Upper triangular matrix → det = product of diagonal = \(1 \times 4 \times 6 = \mathbf{24}\)

Problem 2: If det(A) = 5, find det(3A) where A is a 4×4 matrix.

Solution: \(\det(3A) = 3^4 \cdot \det(A) = 81 \times 5 = \mathbf{405}\)

Problem 3: Find the rank of \(A = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 1 & 3 & 4 \end{pmatrix}\)

Solution:

  1. \(R_2 \leftarrow R_2 - 2R_1\): \(\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 1 & 3 & 4 \end{pmatrix}\)
  2. \(R_3 \leftarrow R_3 - R_1\): \(\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 1 & 1 \end{pmatrix}\)
  3. Non-zero rows = 2 → rank = 2

Hardest GATE Questions

Topic: Determinant Properties

Question (GATE 2019 Style): If A is a 3×3 matrix such that det(A) = 4, find det(2A⁻¹(adj(A))ᵀ).

Solution:

  1. \(\det(adj(A)) = [\det(A)]^{n-1} = 4^2 = 16\)
  2. \(\det((adj(A))^T) = \det(adj(A)) = 16\)
  3. \(\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{4}\)
  4. \(\det(2A^{-1}) = 2^3 \cdot \det(A^{-1}) = 8 \times \frac{1}{4} = 2\)
  5. \(\det(2A^{-1}(adj(A))^T) = \det(2A^{-1}) \cdot \det((adj(A))^T) = 2 \times 16 = \mathbf{32}\)

Why it's hard: Requires combining multiple determinant properties systematically.

Topic: Rank and Nullity

Question (GATE 2017 Variant): Let A be a 6×8 matrix with rank 4. Let B be an 8×6 matrix with rank 3. What is the maximum possible rank of AB?

Solution:

  • AB is a 6×6 matrix
  • rank(AB) ≤ min(rank(A), rank(B)) = min(4, 3) = 3
  • Maximum possible rank = 3

Topic: Singular Matrix Condition

Question: For what value of k is the matrix \(A = \begin{pmatrix} 1 & 2 & k \\ 2 & k & 8 \\ k & 8 & 26 \end{pmatrix}\) singular?

Solution:

  1. Matrix is singular when det(A) = 0

  2. Using cofactor expansion along row 1: \(\det(A) = 1(26k - 64) - 2(52 - 8k) + k(16 - k^2)\)

  3. \(= 26k - 64 - 104 + 16k + 16k - k^3\)

  4. \(= -k^3 + 58k - 168\)
  5. Setting \(-k^3 + 58k - 168 = 0\)
  6. \(k^3 - 58k + 168 = 0\)
  7. Testing k = 4: \(64 - 232 + 168 = 0\) ✓
  8. k = 4 (and two other roots from factoring)

References