List of top Mathematics Questions on Matrix algebra asked in IIT JAM MA

Consider matrices \( P \) of order \( 4 \times 6 \) and \( Q \) of order \( 6 \times 4 \) with real entries such that

Let \( P \) be a \( 4 \times 4 \) matrix with entries from the set of rational numbers. If \( \sqrt{2} + i \), with \( i = \sqrt{-1} \), is a root of the characteristic polynomial of \( P \) and \( I \) is the \( 4 \times 4 \) identity matrix, then

Let \( \alpha, \beta, \gamma, \delta \) be the eigenvalues of the matrix

Then \( \alpha^2 + \beta^2 + \gamma^2 + \delta^2 = \) ..........

Let

Let \( M \) be the matrix whose columns are \( v_1, v_2, 2v_1 - v_2, v_1 + 2v_2 \) in that order. Then the number of linearly independent solutions of the homogeneous system of linear equations \( Mx = 0 \) is ...........

Let

Then \[ \lim_{n \to \infty} M^n x \]

Let \( f_1(x), f_2(x), g_1(x), g_2(x) \) be differentiable functions on \( \mathbb{R} \). Let

Then \( F'(x) \) is equal to

Let \( P \) be a \( 7 \times 7 \) matrix of rank 4 with real entries. Let \( a \in \mathbb{R}^7 \) be a column vector. Then the rank of \( P + aa^T \) is at least ..........

Let \( M \) be an \( n \times n \) matrix with real entries such that \( M^3 = I \). Suppose that \( Mv \neq v \) for any non-zero vector \( v \). Then which of the following statements is/are TRUE?

Let \( \mathcal{M} \) be the set of all invertible \( 5 \times 5 \) matrices with entries 0 and 1. For each \( M \in \mathcal{M} \), let \( n_1(M) \) and \( n_0(M) \) denote the number of 1's and 0's in \( M \), respectively. Then \[ \min_{M \in \mathcal{M}} |n_1(M) - n_0(M)| = \]

Let \( P \) be a \( 7 \times 7 \) matrix of rank 4 with real entries. Let \( a \in \mathbb{R}^7 \) be a column vector. Then the rank of \( P + aa^T \) is at least ..........

Let G be a subgroup of GL₂(ℝ) generated by

\[ \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix} \] Then the order of G is ...........

If \( X \) and \( Y \) are \( n \times n \) matrices with real entries, then which of the following is/are TRUE?

Let \( \mathcal{M} \) be the set of all invertible \( 5 \times 5 \) matrices with entries 0 and 1. For each \( M \in \mathcal{M} \), let \( n_1(M) \) and \( n_0(M) \) denote the number of 1's and 0's in \( M \), respectively. Then \[ \min_{M \in \mathcal{M}} |n_1(M) - n_0(M)| = \]