Linear Algebra - Mathematics Previous Year Questions

Let \(T: \mathbb{R}^4 \to \mathbb{R}^3\) be the linear map satisfying \(T(e_1) = e_1, T(e_2) = e_3, T(e_3) = 0, T(e_4) = e_3\), where \(\{e_1, e_2, e_3, e_4\}\) is the standard basis of \(\mathbb{R}^4\). Then

If a \( 3 \times 3 \) real skew-symmetric matrix has an eigenvalue \( 2i \), then one of the remaining eigenvalues is

Let \( T : \mathbb{C}^3 \to \mathbb{C}^3 \) be defined by \( T \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix} = \begin{pmatrix} z_1 - i z_2 \\ i z_1 + z_3 \\ z_1 + z_2 + i z_3 \end{pmatrix}. \) Then, the adjoint \( T^* \) of \( T \) is given by \( T^* \begin{pmatrix} z_1 \\ z_2 \\ z_3 \end{pmatrix} = \)

Let \( M \) be a \( 3 \times 3 \) real symmetric matrix with eigenvalues \( 0, 2 \) and \( a \) with the respective eigenvectors \( u=(4, b, c)^{T} \), \( v=(-1, 2, 0)^{T} \) and \( w=(1, 1, 1)^{T} \).
Consider the following statements:
I. \( a+b-c=10 \).
II. The vector \( x=\left(0, \frac{3}{2}, \frac{1}{2}\right)^{T} \) satisfies \( M x=v+w \).
III. For any \( d \in \operatorname{span}\{u, v, w\} \), \( M x=d \) has a solution.
IV. The trace of the matrix \( M^{2}+2 M \) is 8.
(\( y^{T} \) denotes the transpose of the vector \( y \))
Which of the above statements are TRUE?

Suppose \(V\) is a finite dimensional vector space over \(\mathbb{R}\). If \(W_1, W_2\) and \(W_3\) are subspaces of \(V\), then which of the following statements is TRUE?

Let \(A\) be a square matrix such that \(\det(xI - A) = x^4(x - 1)^2(x - 2)^3\), where \(\det(M)\) denotes the determinant of a square matrix \(M\).
If \(\text{rank}(A^2) < \text{rank}(A^3) = \text{rank}(A^4)\), then the geometric multiplicity of the eigenvalue 0 of \(A\) is ________.