CSIR NET Maths previous year question papers

Single choice · Vector Spaces and Linear Maps

1. A

   A does not have any non-zero invariant subspace in $\mathbb{C}^{n}$.

2. B

   A has an invariant subspace in $\mathbb{C}^{n}$ of dimension n − 3.

3. C

   All eigenvalues of A are real.

4. D

   $A^{2}$ does not have any invariant subspace in $\mathbb{C}^{n}$ of dimension n − 1.

Single choice · Vector Spaces and Linear Maps

1. A

   The product of two 2×2 real matrices of rank 2 is of rank 2.

2. B

   The product of two 3×3 real matrices of rank 2 is of rank at most 2.

3. C

   The product of two 3×3 real matrices of rank 2 is of rank at least 2.

4. D

   The product of two 2×2 real matrices of rank 1 can be the zero matrix.

Single choice · Vector Spaces and Linear Maps

1. A

   If A is a real 3×2 matrix and B a real 2×3 matrix with BA $= I_{2}$, then AB $= I_{3}$.

2. B

   Let A = [[3, 3], [1, 2]]. Then there is a matrix B with integer entries such that AB $= I_{2}$.

3. C

   Let A = [[3, 1], [1, 2]] with entries in $\mathbb{Z}/6\mathbb{Z}$. Then there is a matrix B with entries in $\mathbb{Z}/6\mathbb{Z}$ such that AB $= I_{2}$.

4. D

   If A is a real non-zero 3×3 diagonal matrix, then there is a real matrix B such that AB $= I_{3}$.

Single choice · Eigenvalues and Canonical Forms

1. A

   1

2. B

   n

3. C

   n(n+1)/2

4. D

   n(n+1)(2n+1)/6

Single choice · Inner Product Spaces and Forms

1. A

   $A^{2} = 0$.

2. B

   rank A = 1.

3. C

   rank A = 0.

4. D

   There exists $u \in \mathbb{R}^{2}, u \ne 0$ such that (u, u) = 0.

Single choice · Inner Product Spaces and Forms

1. A

   A_a is positive definite for all a < 3.

2. B

   A_a is positive definite for all a > 3.

3. C

   A_a is positive definite for all a ≥ −2.

4. D

   A_a is positive definite only for finitely many values of a.