IIT JAM - Mathematical Statistics (MS) · India · IITs (JAM 2026 organised by IIT Bombay)
Original practice questions written to the published syllabus · not past papersVector 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)
How our questions for IIT JAM spread across the units of Linear Algebra & Matrix Theory for Statistics. Difficulty: 3 easy, 22 medium, 4 hard.
| Vector spaces, subspaces, linear independence, basis and dimension | 3 | |
| Determinants, inverses and partitioned matrices | 3 | |
| Systems of linear equations and their consistency | 3 | |
| Eigenvalues, eigenvectors and the Cayley-Hamilton theorem | 3 | |
| Idempotent and projection matrices | 3 | |
| Kronecker products and vec operators | 3 | |
| Rank, nullity and row-reduced echelon form | 2 | |
| Inner products, orthogonality and Gram-Schmidt orthonormalisation | 2 | |
| Diagonalisation and spectral decomposition of symmetric matrices | 2 | |
| Quadratic forms, positive definite and semi-definite matrices | 2 | |
| Generalised inverses (g-inverse and Moore-Penrose inverse) | 2 | |
| Singular value decomposition | 1 |
Choose an answer, then open the working. Each explanation says why the right answer is right and why the others are not.
What is the dimension of the space of all 3 × 3 real symmetric matrices?
B. 6
n(n + 1)/2 = 6 free entries (diagonal plus upper triangle). 3 is the skew-symmetric dimension.
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)
A system of 3 equations in 3 unknowns has rank(A) = rank([A | b]) = 2. The solution set is:
C. A line (one free parameter)
Solution dimension = n − rank = 3 − 2 = 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)
For which value of k does x + y = 2, 2x + 2y = k have infinitely many solutions?
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)
A 3 × 3 matrix has eigenvalues 1, 2 and 3. What is det(A⁻¹)?
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)
Using Cayley–Hamilton, the inverse of A with rows (1, 2) and (3, 4) is:
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)
Which matrix is not diagonalisable?
C. Rows (1, 1) and (0, 1)
It has the repeated eigenvalue 1 with only one independent eigenvector.
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)
The quadratic form x² + 4xy + y² is:
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)
The eigenvalues of an idempotent matrix are:
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)
The Moore–Penrose inverse of the column vector a = (1, 2, 2)′ is:
B. (1, 2, 2)/9
a⁺ = a′/(a′a) and a′a = 9.
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)
If rank A = 2 and rank B = 3, rank(A ⊗ B) is:
B. 6
rank(A ⊗ B) = rank A × rank B = 6; 5 adds the ranks.
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)
Ten here, 29 in the app. The Laisha app has every practice question we have written to IIT JAM in this subject, marked as you go, with timed papers, a jump-to-unanswered list and a certificate when you pass at 70 per cent.
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