List of top Linear Algebra Questions asked in IIT JAM MA

For a 4 x 4 matrix \(M\isin M_4(\mathbb{C})\), let M denote the matrix obtained from M by replacing each entry of M by its complex conjugate. Consider the real vector space
H = \(\{M\isin M_4(\mathbb{C}):M^T=\overline M\)
where M^T denotes the transpose of M. The dimension of H as a vector space over \(\R\) is equal to

Let be a finite group of order at least two and let e denote the identity element of G. Let σ: G →g be a bijective group homomorphism that satisfies the following two conditions:
(i): if σ(g)=g for some gεG, then g=e,
(ii) (σ o σ) (g)=g for all g σ G.
The n which of the following is/are correct ?

Let P be a fixed 3×3 matrix with entries in \(\R\). Which of the following maps from \(M_3(\R)\) to \(M_3(\R)\) is/are linear?

The number of distinct subgroup of Z₉₉₉ is_____.

The number of elements of order 12 in the symmetric group S₇ is equal to____

Let \(A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ -1 & 1 \end{pmatrix}\)and let A^T denote the transpose of A. Let \(u=\begin{pmatrix} u_1 \\ u_2 \end{pmatrix}\) and \(v=\begin{pmatrix} v_1 \\ v_2 \\ v_3 \end{pmatrix}\)be column vectors with entries in \(\R\) such that \(u^2_1+u_2^2=1\) and \(v_1^2+v_2^2+v^2_3=1\). Suppose
\(Au=\sqrt2v\) and \(A^Tv=\sqrt2u.\)
Then |\(𝑢1 + 2 \sqrt2 𝑣_1\)| is equal to _________. (Rounded off to two decimal places)

Consider the 4 × 4 matrix \(M=\begin{pmatrix} 11 & 10 & 10 & 10 \\ 10 & 11 & 10 & 10 \\ 10 & 10 & 11 & 10 \\ 10 & 10 & 10 & 11 \end{pmatrix}\). Then the value of the determinant of M is equal to _________.

Let σ be the permutation in the symmetric group S₅ given by
σ(1) = 2, σ(2) = 3, σ(3) = 1, σ(4) = 5, σ(5) = 4.
Define
N(σ) = {𝜏 ∈ S₅ ∶ σ ∘ 𝜏 = 𝜏 ∘ σ}.
Then the number of elements in N(σ) is equal to _________.

Let \( G \) be a group with identity \( e \). Let \( H \) be an abelian non-trivial proper subgroup of \( G \) with the property that \( H \cap gHg^{-1} = \{ e \} \) for all \( g \notin H \). If \( K = \{ g \in G : gh = hg \text{ for all } h \in H \}, \) then

Let \( F = \{\omega \in \mathbb{C} : \omega^{2020} = 1\}. \)

Consider the groups \[ G = \left\{ \begin{pmatrix} \omega & z \\ 0 & 1 \end{pmatrix} : \omega \in F, z \in \mathbb{C} \right\} \text{and} H = \left\{ \begin{pmatrix} 1 & z \\ 0 & 1 \end{pmatrix} : z \in \mathbb{C} \right\} \] under matrix multiplication.

Then the number of cosets of \( H \) in \( G \) is

Let \( M \) be a \( 4 \times 3 \) real matrix and let \( \{e_1, e_2, e_3\} \) be the standard basis of \( \mathbb{R}^3. \) Which of the following is true?

Let \( M \) be an \( n \times n \) ( \( n \ge 2 \) ) non-zero real matrix with \( M^2 = 0 \) and let \( \alpha \in \mathbb{R} \setminus \{0\}. \) Then

Let \( M \) be a real \( 6 \times 6 \) matrix. Let 2 and -1 be two eigenvalues of \( M. \) If \( M^5 = aI + bM \), where \( a, b \in \mathbb{R}, \) then

Consider the following group under matrix multiplication:

\[ H = \left\{ \begin{bmatrix} 1 & p & q \\ 0 & 1 & r \\ 0 & 0 & 1 \end{bmatrix} : p, q, r \in \mathbb{R} \right\}. \]

Then the center of the group is isomorphic to

Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be the linear transformation given by \( T(x, y) = (-x, y). \) Then

Let \( T : \mathbb{R}^7 \to \mathbb{R}^7 \) be a linear transformation with \( \text{Nullity}(T) = 2. \) Then, the minimum possible value for \( \text{Rank}(T^2) \) is ............

Let \( S^1 = \{z \in \mathbb{C} : |z| = 1\} \) be the circle group under multiplication and \( i = \sqrt{-1}. \) Then the set \( \{\theta \in \mathbb{R} : (e^{i2\pi\theta}) \text{ is infinite}\} \) is

Suppose that \( G \) is a group of order 57 which is not cyclic. If \( G \) contains a unique subgroup \( H \) of order 19, then for any \( g \notin H \), the order of \( g \) is ................

Let \[ M = \begin{bmatrix} 9 & 2 & 7 & 1 \\ 0 & 7 & 2 & 1 \\ 0 & 0 & 11 & 6 \\ 0 & 0 & -5 & 0 \end{bmatrix}. \] Then, the value of \( \det((8I - M)^3) \) is ................

Let \( \phi : S_3 \to S_1 \) be a non-trivial non-injective group homomorphism. Then the number of elements in the kernel of \( \phi \) is .............

Consider the real vector space \( P_{2020} = \left\{ \sum_{i=0}^n a_i x^i : a_i \in \mathbb{R}, \, 0 \le n \le 2020 \right\}. \) Let \( W \) be the subspace given by \[ W = \left\{ \sum_{i=0}^n a_i x^i \in P_{2020} : a_i = 0 \text{ for all odd } i \right\}. \] Then the dimension of \( W \) is ................

Let \( V \) be a non-zero vector space over a field \( F \). Let \( S \subset V \) be a non-empty set. Consider the following properties of \( S \):
(I) For any vector space \( W \) over \( F \), any map \( f : S \to W \) extends to a linear map from \( V \) to \( W \).
(II) For any vector space \( W \) over \( F \) and any two linear maps \( f, g : V \to W \) satisfying \( f(s) = g(s) \) for all \( s \in S \), we have \( f(v) = g(v) \) for all \( v \in V \).
(III) \( S \) is linearly independent.
(IV) The span of \( S \) is \( V \).
Which of the following statement(s) is/are true?