Matrices, Determinants & Eigenvalues

Engineering Mathematics carries ~13 marks in GATE ME, and Linear Algebra reliably supplies 2–3 of them. Eigenvalues are the single most-asked idea here — usually a fast 1-mark NAT if you know the shortcuts. Treat this as guaranteed, low-effort marks.

Core results

For an n×n matrix A: det(A) = product of eigenvalues; trace(A) = sum of eigenvalues. Eigenvalues λ solve the characteristic equation det(A − λI) = 0. Rank = number of non-zero rows in row-echelon form = number of non-zero eigenvalues only for diagonalisable/symmetric cases (rank counts non-zero singular values in general).

System Ax = b: unique solution if rank(A) = rank([A|b]) = n; infinite solutions if rank(A) = rank([A|b]) < n; no solution if rank(A) ≠ rank([A|b]). Homogeneous Ax = 0 has non-trivial solutions only when det(A) = 0.

Special matrices: eigenvalues of a triangular/diagonal matrix are its diagonal entries; symmetric real matrices have real eigenvalues and orthogonal eigenvectors; a matrix and its transpose share eigenvalues.

Exam Tricks & Tips

• 🎯 Never expand the characteristic polynomial for 2×2: λ² − (trace)λ + det = 0 gives both roots instantly.
• 🎯 Sum-and-product check: if λ₁+λ₂+λ₃ ≠ trace or λ₁λ₂λ₃ ≠ det, you slipped — recompute before answering.
• 🎯 Eigenvalues of A⁻¹ are 1/λ; of Aⁿ are λⁿ; of (A + kI) are (λ + k) — same eigenvectors. Most "find eigenvalue of A² + 3A" questions are just this.
• 🎯 Cayley–Hamilton lets you replace Aⁿ by lower powers — the fastest route for "compute A⁵" or A⁻¹ in terms of A.
• 🎯 Rank shortcut: if any row is a scalar multiple/linear combo of others, det = 0 and rank drops — spot it before doing full elimination.
• ❌ Common mistake: assuming eigenvectors are unique — they are defined only up to a non-zero scalar, and repeated eigenvalues may not give enough independent eigenvectors (defective matrix).

Expected exam pattern

One 1-mark NAT on eigenvalues (often using the A⁻¹/Aⁿ shortcut) plus a 2-mark question on rank / consistency of a linear system or on Cayley–Hamilton. Occasionally a property-based MSQ (which statements about symmetric/orthogonal matrices are true).

Quick recap

Trace = Σλ, det = Πλ. Solve det(A − λI) = 0; use λ→1/λ, λⁿ, λ+k transforms without recomputing. Consistency from rank comparison. Cayley–Hamilton kills high powers. Watch defective (repeated-eigenvalue) matrices.

Cover the answer, recall, then check. 12 cards on linear algebra for GATE ME.

Q1. How do trace and determinant relate to eigenvalues?
A1. trace(A) = sum of all eigenvalues; det(A) = product of all eigenvalues. Works for any square matrix.

Q2. What equation gives the eigenvalues of A?
A2. The characteristic equation det(A − λI) = 0. For 2×2: λ² − trace·λ + det = 0.

Q3. If λ is an eigenvalue of A, what are the eigenvalues of A⁻¹, Aⁿ, and A + kI?
A3. 1/λ, λⁿ, and λ + k respectively — all with the SAME eigenvector.

Q4. State the condition for Ax = b to have a unique solution.
A4. rank(A) = rank([A|b]) = n (number of unknowns). If equal but < n → infinite solutions; if unequal → no solution.

Q5. When does the homogeneous system Ax = 0 have a non-trivial solution?
A5. Only when det(A) = 0 (A is singular), i.e. rank(A) < n.

Q6. What are the eigenvalues of a triangular or diagonal matrix?
A6. Its diagonal entries, directly — no computation needed.

Q7. State the Cayley–Hamilton theorem and its main use.
A7. Every matrix satisfies its own characteristic equation. Use it to express high powers Aⁿ (or A⁻¹) in terms of lower powers of A.

Q8. What special properties do real symmetric matrices have?
A8. All eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal; always diagonalisable.

Q9. How are eigenvalues of A and Aᵀ related?
A9. They are identical (A and Aᵀ have the same characteristic polynomial), though eigenvectors generally differ.

Q10. What is a defective matrix?
A10. A matrix with a repeated eigenvalue that lacks a full set of independent eigenvectors, so it cannot be diagonalised.

Q11. For an orthogonal matrix Q, what is Q⁻¹ and det(Q)?
A11. Q⁻¹ = Qᵀ, and det(Q) = ±1. Its eigenvalues have magnitude 1.

Q12. If A is 3×3 with eigenvalues 1, 2, 3, find det(A² + A).
A12. Eigenvalues of A²+A are λ²+λ: 2, 6, 12. det = 2·6·12 = 144.

Almost every "Engineering Mathematics" block in GATE ME opens with a matrix question, and it is free marks if your algebra is reflexive. Eigenvalues alone appear in vibrations (natural frequencies), principal stresses (Mohr's circle), and stability — so this one topic quietly pays off across the whole paper.

Core concept: a square matrix is a linear operator; its determinant measures how it scales volume (and whether it is invertible), and its eigenvalues are the special scaling factors along directions the matrix does not rotate.

Deep explanation

Beginner — determinants and rank

For a 2×2, det = ad − bc. For 3×3 expand along any row/column using cofactors. Key facts you must reflex:

• det(AB) = det(A)·det(B); det(Aᵀ) = det(A); det(A⁻¹) = 1/det(A).
• A is invertible ⇔ det(A) ≠ 0 ⇔ full rank ⇔ rows/columns linearly independent.
• Rank = number of non-zero rows after row-reduction = size of the largest non-zero minor. It fixes how many independent equations you truly have.

For a system Ax = b: unique solution if rank(A) = rank([A|b]) = n; infinite solutions if the common rank < n; no solution if rank(A) < rank([A|b]).

Intermediate — eigenvalues and eigenvectors

λ is an eigenvalue of A if A x = λ x for some non-zero x. Solve the characteristic equation det(A − λI) = 0. Two shortcuts that save enormous time:

• Sum of eigenvalues = trace (sum of diagonal).
• Product of eigenvalues = det(A).

So for a 2×2 you can often write λ² − (trace)λ + det = 0 directly. Eigenvectors come from solving (A − λI)x = 0 for each λ.

Advanced — properties GATE loves to test

• Eigenvalues of a triangular/diagonal matrix are its diagonal entries.
• Eigenvalues of Aⁿ are λⁿ; of A⁻¹ are 1/λ; of A + kI are λ + k. The eigenvectors stay the same.
• Symmetric real matrices have real eigenvalues and orthogonal eigenvectors (this is why principal stresses are always real).
• Cayley–Hamilton: every matrix satisfies its own characteristic equation — use it to compute A⁻¹ or high powers without brute force.

Worked example

Find the eigenvalues of A = [[4, 1],[2, 3]].
Trace = 4 + 3 = 7, det = 4·3 − 1·2 = 10. Characteristic equation: λ² − 7λ + 10 = 0 → (λ − 5)(λ − 2) = 0. λ = 5, 2. Check: 5 + 2 = 7 ✓ and 5·2 = 10 ✓. No cofactor expansion needed.

GATE relevance

Expect 1–2 marks on eigenvalues/rank almost every year, plus hidden use in TOM (whirling/natural frequency), SOM (principal stress = eigenvalue of stress tensor), and control-like stability arguments. It is high-yield, low-effort.

Exam tricks & shortcuts

• Always test with trace = Σλ, det = Πλ before trusting a computed eigenvalue.
• If asked for eigenvalues of A⁻¹ or A³, do NOT recompute — transform λ.
• Mnemonic: "Trace sums, Determinant multiplies."

Key formulas

• Inverse: A⁻¹ = adj(A)/|A| (|A| ≠ 0); |AB| = |A||B|.
• Rank & consistency: system consistent if rank(A) = rank(A|B); unique solution ⇔ rank = number of unknowns.
• Eigenvalues: |A − λI| = 0; Σλ = trace(A), Πλ = |A|.
• Properties: eigenvalues of A⁻¹ are 1/λ, of Aⁿ are λⁿ.
• Cayley–Hamilton: A satisfies its characteristic equation.

Eigenvalues characterise principal directions; rank determines solvability of linear systems.

Worked Example

Problem: Find all eigenvalues of the symmetric matrix
A = [[2, 1, 1], [1, 2, 1], [1, 1, 2]].

Solution:
The eigenvalues satisfy det(A − λI) = 0. Expanding the characteristic determinant gives the cubic:
λ³ − 6λ² + 9λ − 4 = 0.
(The coefficient 6 is the trace 2+2+2, and the constant 4 is det A.)

Look for a rational root. Try λ = 1:
1 − 6 + 9 − 4 = 0 ✓, so (λ − 1) is a factor.

Divide out (λ − 1):
λ³ − 6λ² + 9λ − 4 = (λ − 1)(λ² − 5λ + 4).
Factor the quadratic: λ² − 5λ + 4 = (λ − 1)(λ − 4).

Hence the eigenvalues are λ = 1 (repeated) and λ = 4.

Verification: sum of eigenvalues = 1 + 1 + 4 = 6 = trace(A) ✓; product = 1 × 1 × 4 = 4 = det(A) ✓.

Answer: λ = 1, 1, 4.