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GATE Linear Algebra & Matrix Theory for Statistics: practice questions

GATE - Statistics (ST) · India · IITs and IISc on rotation (2026: IIT Guwahati)

Original practice questions written to the published syllabus · not past papers
10worked here, free
37written to this exam in this subject
12syllabus units

What this subject covers

Vector spaces, rank, inverses, eigenvalues, quadratic forms and generalised inverses: the language of linear models, multivariate analysis and mixed models. Taught here with the statistician's applications (projection matrices, idempotence, the distribution of quadratic forms).

In the exams: GATE ST 'Matrix Theory' (heavy); GATE DA linear algebra (heavy); IIT JAM MS (heavy); AIEEA-PG Group 07 Unit II (present); AICE-JRF Ag Stat Unit 4 linear models (assumed)

Where the 37 questions fall

How our questions for GATE spread across the units of Linear Algebra & Matrix Theory for Statistics. Difficulty: 5 easy, 26 medium, 6 hard.

Systems of linear equations and their consistency4
Eigenvalues, eigenvectors and the Cayley-Hamilton theorem4
Generalised inverses (g-inverse and Moore-Penrose inverse)4
Kronecker products and vec operators4
Rank, nullity and row-reduced echelon form3
Determinants, inverses and partitioned matrices3
Diagonalisation and spectral decomposition of symmetric matrices3
Idempotent and projection matrices3
Singular value decomposition3
Vector spaces, subspaces, linear independence, basis and dimension2
Inner products, orthogonality and Gram-Schmidt orthonormalisation2
Quadratic forms, positive definite and semi-definite matrices2

10 practice questions, worked

Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.

1 · Rank, nullity and row-reduced echelon form · medium

Multiplying a matrix A on the left by a non-singular matrix P:

  1. Always increases rank
  2. Always reduces rank by one
  3. Leaves rank unchanged
  4. Makes the product symmetric
Answer and working

C. Leaves rank unchanged

Invertible matrices preserve rank: rank(PA) = rank(A).

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

2 · Systems of linear equations and their consistency · easy

For which value of k does x + y = 2, 2x + 2y = k have infinitely many solutions?

  1. k = 2
  2. k = 0
  3. k = 4
  4. No value of k
Answer and working

C. k = 4

The second equation must be twice the first; for any other k the system is inconsistent.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

3 · Eigenvalues, eigenvectors and the Cayley-Hamilton theorem · medium

A 3 × 3 matrix has eigenvalues 1, 2 and 3. What is det(A⁻¹)?

  1. 6
  2. 1/3
  3. 11/6
  4. 1/6
Answer and working

D. 1/6

det A = 1 × 2 × 3 = 6, so det(A⁻¹) = 1/6. 11/6 is the sum of the reciprocal eigenvalues, the trace of A⁻¹.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

4 · Eigenvalues, eigenvectors and the Cayley-Hamilton theorem · medium

For A with rows (1, 2) and (3, 4), Cayley–Hamilton gives A² equal to:

  1. 5A − 2I
  2. 4A + I
  3. A + 2I
  4. 5A + 2I
Answer and working

D. 5A + 2I

Characteristic polynomial λ² − 5λ − 2, so A² − 5A − 2I = 0.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

5 · Eigenvalues, eigenvectors and the Cayley-Hamilton theorem · hard

Using Cayley–Hamilton, the inverse of A with rows (1, 2) and (3, 4) is:

  1. (5I − A)/2
  2. (A + 5I)/2
  3. 2(A − 5I)
  4. (A − 5I)/2
Answer and working

D. (A − 5I)/2

From A² − 5A = 2I, A(A − 5I) = 2I.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

6 · Quadratic forms, positive definite and semi-definite matrices · medium

The quadratic form x² + 4xy + y² is:

  1. Positive definite
  2. Indefinite
  3. Negative definite
  4. Positive semi-definite
Answer and working

B. Indefinite

Eigenvalues of rows (1, 2), (2, 1) are 3 and −1.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

7 · Idempotent and projection matrices · medium

The eigenvalues of an idempotent matrix are:

  1. −1 or 1
  2. Any real numbers
  3. 0 or 1
  4. All equal to the trace
Answer and working

C. 0 or 1

λ² = λ gives λ = 0 or 1.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

8 · Generalised inverses (g-inverse and Moore-Penrose inverse) · easy

A g-inverse G of a matrix A is defined by:

  1. AGA = A
  2. GA = I
  3. AG = I
  4. G = A′
Answer and working

A. AGA = A

Only AGA = A is required; G need not give the identity.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

9 · Kronecker products and vec operators · easy

If A is 2 × 3 and B is 4 × 5, the Kronecker product A ⊗ B is of order:

  1. 8 × 15
  2. 6 × 20
  3. 2 × 5
  4. 8 × 8
Answer and working

A. 8 × 15

(mp) × (nq) = (2 × 4) × (3 × 5).

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

10 · Kronecker products and vec operators · medium

The eigenvalues of A ⊗ B are:

  1. All sums λᵢ + μⱼ
  2. Only the eigenvalues of A
  3. The squares of the eigenvalues of B
  4. All products λᵢμⱼ
Answer and working

D. All products λᵢμⱼ

Sums appear for the Kronecker sum A ⊗ I + I ⊗ B, not the product.

Further reading: Searle and Khuri, Matrix Algebra Useful for Statistics (2017); Harville, Matrix Algebra From a Statistician's Perspective (1997); Strang, Introduction to Linear Algebra (2016); Rao, Linear Statistical Inference (1973)

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Linear Algebra & Matrix Theory for StatisticsThe full study manual, free GATEFormat, sections and how it has changed

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