List of top Mathematics Questions on Matrix algebra

Consider the real matrices
Let \(A\) be an \(m\times n\) real matrix with non-zero entries. Which of the following statements is/are true?
Let \(A\) be an \(n\times n\) real matrix. Consider the following statements.
Consider matrices \( P \) of order \( 4 \times 6 \) and \( Q \) of order \( 6 \times 4 \) with real entries such that

Find the rank of the matrix: \[ \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 2 \\ 2 & 5 & 6 & 4 \\ 2 & 6 & 8 & 5 \end{bmatrix} \] Rank = ?

For real constants $a$ and $b$, let \[ M = \begin{bmatrix} \dfrac{1}{\sqrt{2}} & \dfrac{1}{\sqrt{2}} \\ a & b \end{bmatrix} \] be an orthogonal matrix. Then which of the following statements is/are always TRUE?

Let $M$ be an $n \times n$ non-zero skew symmetric matrix. Then the matrix $(I_n - M)(I_n + M)^{-1}$ is always
 

Let $T: \mathbb{R}^3 \to \mathbb{R}^4$ be a linear transformation. If $T(1,1,0) = (2,0,0,0)$, $T(1,0,1) = (2,4,0,0)$, and $T(0,1,1) = (0,0,2,0)$, then $T(1,1,1)$ equals
 

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 \( M = \begin{bmatrix} \tfrac{1}{4} & \tfrac{3}{4} \\ \\ \tfrac{3}{5} & \tfrac{2}{5} \end{bmatrix}. \) If \( I \) is the \( 2 \times 2 \) identity matrix and \( 0 \) is the \( 2 \times 2 \) zero matrix, then

Let \( M \) be a \( 3 \times 3 \) non-zero, skew-symmetric real matrix. If \( I \) is the \( 3 \times 3 \) identity matrix, then

Let \( M = \sum_{i=1}^{4} X_i X_i^T \), where \[ X_1^T = [1 \ -1 \ 1 \ 0], X_2^T = [1 \ 1 \ 0 \ 1], X_3^T = [1 \ 3 \ 1 \ 0] \, \text{and} \, X_4^T = [1 \ 1 \ 1 \ 0]. \] Then the rank of \( M \) equals ............... 
 

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 \[ \min_{M \in \mathcal{M}} |n_1(M) - n_0(M)| = \]