Matrices and Determinants

Matrices and determinants are the entry point to GATE EC Engineering Mathematics (which carries ~13 marks / paper). They power circuit-equation solving (nodal/mesh), state-space, and appear as direct 1–2 mark questions on rank, determinant value, and inverse.

Key results

• det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(A⁻¹) = 1/det(A); det(kA) = kⁿ·det(A) for n×n.
• A is invertible ⇔ det(A) ≠ 0 ⇔ rank = n ⇔ columns independent.
• (AB)⁻¹ = B⁻¹A⁻¹; (AB)ᵀ = BᵀAᵀ.
• Rank = number of non-zero rows in row-echelon form = order of the largest non-zero minor.
• A⁻¹ = adj(A)/det(A); adj(A) = transpose of cofactor matrix.

Matrix type	Defining property	Determinant
Symmetric	A = Aᵀ	real eigenvalues
Skew-symmetric (odd n)	A = −Aᵀ	0
Orthogonal	AᵀA = I	±1
Triangular	zeros one side	product of diagonal
Singular	det = 0	not invertible

Exam Tricks & Tips

• 🎯 Determinant = product of eigenvalues; trace = sum of eigenvalues — use to cross-check without expansion.
• 🎯 For a triangular (or diagonal) matrix, det is just the diagonal product — spot this instantly.
• 🎯 Row/column operations don't change rank; swapping two rows flips det sign, scaling a row by k scales det by k.
• 🎯 Skew-symmetric matrix of odd order always has det = 0 (a common trap answer).
• 🎯 For 2×2, A⁻¹ = (1/det)·[[d,−b],[−c,a]] — memorise the swap-and-negate pattern.
• 🎯 Rank of a product: rank(AB) ≤ min(rank A, rank B).
• ❌ Common mistake: assuming det(A+B) = det(A)+det(B) — determinant is NOT additive; only multiplicative.

Expected exam pattern

1-mark: find det/rank/inverse of a small numeric matrix, or identify singular condition (solve for a parameter making det = 0). 2-mark: rank after operations, or det via eigenvalue/trace shortcuts.

Quick recap

Determinant is multiplicative not additive; det ≠ 0 ⇔ full rank ⇔ invertible. Rank = largest non-zero minor. Use trace/det = sum/product of eigenvalues to verify. Skew-symmetric odd order → det 0.

Cover the answer, recall, then check. 12 cards on matrices and determinants for GATE EC.

Q1. When is a square matrix A invertible?
A1. iff det(A) ≠ 0, equivalently rank(A) = n, equivalently its columns (rows) are linearly independent.

Q2. State det(AB) and det(kA) for an n×n matrix.
A2. det(AB) = det(A)·det(B); det(kA) = kⁿ·det(A).

Q3. How is det related to eigenvalues and trace?
A3. det(A) = product of all eigenvalues; trace(A) = sum of diagonal = sum of eigenvalues.

Q4. What is the determinant of any odd-order skew-symmetric matrix?
A4. Zero. (Because det(A) = det(−Aᵀ) = (−1)ⁿ det(A), and for odd n this forces det = 0.)

Q5. Give the inverse formula for a general non-singular matrix.
A5. A⁻¹ = adj(A)/det(A), where adj(A) is the transpose of the cofactor matrix.

Q6. How do you find the rank of a matrix?
A6. Reduce to row-echelon form; rank = number of non-zero rows = order of the largest non-vanishing minor.

Q7. What is det of a triangular or diagonal matrix?
A7. The product of its diagonal entries.

Q8. State (AB)⁻¹ and (AB)ᵀ.
A8. (AB)⁻¹ = B⁻¹A⁻¹ and (AB)ᵀ = BᵀAᵀ (order reverses).

Q9. For a 2×2 matrix [[a,b],[c,d]], write the inverse.
A9. (1/(ad−bc))·[[d,−b],[−c,a]].

Q10. How does swapping two rows affect the determinant?
A10. It multiplies the determinant by −1 (sign flip). Adding a multiple of one row to another leaves det unchanged.

Q11. What property makes AᵀA = I?
A11. A is orthogonal; then det(A) = ±1 and A⁻¹ = Aᵀ.

Q12. Bound on rank of a product AB.
A12. rank(AB) ≤ min(rank(A), rank(B)).

Matrices are the language linear systems, circuits, and transforms all speak, and the determinant is the single number that tells you whether a system has a unique solution, whether a matrix is invertible, and how it scales volume. GATE Engineering Mathematics reliably delivers 1–2 marks here, often as a quick rank or determinant computation that gates a larger problem.

Core concept: A matrix encodes a linear system; its determinant tests invertibility, and its rank counts independent equations, deciding whether solutions are unique, infinite, or absent.

The theory

Beginner — determinant essentials

For A (n×n), det(A) = 0 ⇔ A is singular (non-invertible, rows/columns linearly dependent). Useful properties:

• det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(A^(−1)) = 1/det(A).
• det(kA) = kⁿ·det(A) for an n×n matrix.
• Swapping two rows flips the sign; adding a multiple of one row to another leaves det unchanged.
• Triangular/diagonal matrix: det = product of diagonal entries.

Intermediate — rank and solvability

The rank is the number of linearly independent rows (= independent columns), found by row-reducing to echelon form and counting non-zero rows. For A·x = b with A being m×n:

• Unique solution: rank(A) = rank([A|b]) = n.
• Infinite solutions: rank(A) = rank([A|b]) < n (free variables = n − rank).
• No solution: rank(A) < rank([A|b]) (inconsistent).
  Homogeneous A·x = 0 has a non-trivial solution ⇔ det(A) = 0 (rank < n).

Advanced — inverse and special forms

• Inverse: A^(−1) = adj(A)/det(A); adj(A) is the transpose of the cofactor matrix.
• Cayley-Hamilton: every matrix satisfies its own characteristic equation — a fast route to A^(−1) and high powers of A.
• Special matrices: orthogonal (AᵀA = I, det = ±1), symmetric (A = Aᵀ), skew-symmetric (Aᵀ = −A, so det = 0 for odd n), unitary (complex analogue of orthogonal).
• Trace = sum of diagonal = sum of eigenvalues; det = product of eigenvalues.

Worked example

For what value of k does the system x + 2y + 3z = 0, 2x + ky + 6z = 0, 3x + 6y + 9z = 0 have non-trivial solutions?
Non-trivial ⇔ det = 0. Note row3 = 3·row1, so the matrix is already singular for any k — det = 0 always. So non-trivial solutions exist for all k. (If instead only rows 1,2 were given independently, solve det = 0.)
Sanity: rank ≤ 2 < 3 always ⇒ at least one free variable. ✔

GATE relevance

1–2 mark NAT/MCQ: evaluate a determinant, find rank, decide consistency/number of solutions, compute an inverse, or use det properties (det(kA), det(AB)). Also the entry point to eigenvalue problems and state-space controllability.

Exam tricks and shortcuts

• Spot dependent rows/columns instantly ⇒ det = 0, skip expansion.
• det(kA) = kⁿ det(A) — the exponent n trips people up.
• Product of eigenvalues = det; sum = trace — cross-check without solving.
• Mnemonic: "Rank matches ⇒ consistent; equals n ⇒ unique."

Key formulas

• Determinant: |A| ≠ 0 ⇔ invertible; A⁻¹ = adj(A)/|A|.
• Properties: |AB| = |A||B|; |kA| = kⁿ|A|; |Aᵀ| = |A|.
• Rank: number of linearly independent rows/columns; system consistent if rank(A) = rank(A|B).
• Unique solution ⇔ rank = number of unknowns; infinite if rank < unknowns; none if ranks differ.
• Cayley–Hamilton: every matrix satisfies its own characteristic equation.

Rank determines the solvability of AX = B; Cayley–Hamilton lets you compute powers/inverses of a matrix.

Worked Example

Problem: Using Cramer's rule, solve for x in the linear system
x + y + z = 6, x + 2y + 3z = 14, x + 4y + 9z = 36.

Solution:
Write the coefficient matrix and its determinant D:
A = [[1, 1, 1], [1, 2, 3], [1, 4, 9]].
The nodes 1, 2, 3 form a Vandermonde structure, so
D = (2−1)(3−1)(3−2) = 1 × 2 × 1 = 2.
Since D = 2 ≠ 0, the system has a unique solution and Cramer's rule applies.

For x, replace the first column of A with the right-hand side (6, 14, 36):
Dx = |[[6, 1, 1], [14, 2, 3], [36, 4, 9]]|.
Expand along row 1:
Dx = 6(2×9 − 3×4) − 1(14×9 − 3×36) + 1(14×4 − 2×36)
Dx = 6(18 − 12) − 1(126 − 108) + 1(56 − 72)
Dx = 6(6) − 1(18) + 1(−16) = 36 − 18 − 16 = 2.

Therefore x = Dx / D = 2 / 2 = 1.
Back-substituting confirms the full solution (x, y, z) = (1, 2, 3):
check row 3 → 1 + 4(2) + 9(3) = 1 + 8 + 27 = 36 ✓.

Answer: x = 1 (complete solution (1, 2, 3)).