List of top Matrix Theory Questions

Let \(M\) be any \(3\times3\) symmetric matrix with eigenvalues \(1,2,3\). Let \(N\) be any \(3\times3\) matrix with real eigenvalues such that \(MN+N^{T}M=3I\), where \(I\) is the \(3\times3\) identity matrix. Which of the following cannot be eigenvalue(s) of the matrix \(N\)?

Let \( M \) be a \( 3 \times 2 \) real matrix having a singular value decomposition as \( M = U S V^T \), where the matrix \( S = \begin{bmatrix} \sqrt{3} & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}^T \), \( U \) is a \( 3 \times 3 \) orthogonal matrix, and \( V \) is a \( 2 \times 2 \) orthogonal matrix. Then which of the following statements is/are true?

Let \( M \) be a 3 \(\times\) 3 real symmetric matrix with eigenvalues \(-1, 1, 2\) and the corresponding unit eigenvectors \( u, v, w \), respectively. Let \( x \) and \( y \) be two vectors in \( \mathbb{R}^3 \) such that \[ Mx = u + 2(v + w) \quad \text{and} \quad M^2 y = u - (v + 2w). \] Considering the usual inner product in \( \mathbb{R}^3 \), the value of \( |x + y|^2 \), where \( |x + y| \) is the length of the vector \( x + y \), is

Let \( M \) be any square matrix of arbitrary order \( n \) such that \( M^2 = 0 \) and the nullity of \( M \) is 6. Then the maximum possible value of \( n \) (in integer) is ________

Consider the usual inner product in \( \mathbb{R}^4 \). Let \( u \in \mathbb{R}^4 \) be a unit vector orthogonal to the subspace \[ S = \{(x_1, x_2, x_3, x_4)^T \in \mathbb{R}^4 \mid x_1 + x_2 + x_3 + x_4 = 0 \}. \] If \( v = (1, -2, 1, 1)^T \), and the vectors \( u \) and \( v - \alpha u \) are orthogonal, then the value of \( \alpha^2 \) (rounded off to two decimal places) is equal to ________

Let \(M\) be a 2 \(\times\) 2 real matrix such that \((I + M)^{-1} = I - \alpha M\), where \(\alpha\) is a non-zero real number and \(I\) is the 2 \(\times\) 2 identity matrix. If the trace of the matrix \(M\) is 3, then the value of \(\alpha\) is

Let

\[ A = \begin{bmatrix} a & u_1 & u_2 & u_3 \end{bmatrix}, \quad B = \begin{bmatrix} b & u_1 & u_2 & u_3 \end{bmatrix}, \quad C = \begin{bmatrix} u_2 & u_3 & u_1 & a + b \end{bmatrix}. \]

Let \( \det(A), \det(B), \det(C) \) denote the determinants of the matrices \( A \), \( B \), and \( C \), respectively. If

\[ \det(A) = 6, \quad \det(B) = 2, \quad \text{then } \det(A + B) - \det(C) = \underline{\hspace{2cm}}. \]

Let \( (Y, X_1, X_2) \) be a random vector with mean vector

\[ \begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix} \]

and covariance matrix

\[ \begin{pmatrix} 10 & 0.5 & -0.5 \\ 0.5 & 7 & 1.5 \\ -0.5 & 1.5 & 2 \end{pmatrix} \]

Then the value of the multiple correlation coefficient between \( Y \) and its best linear predictor on \( X_1 \) and \( X_2 \) equals _________ (round off to 2 decimal places).

Let \( A \) be the 2 × 2 real matrix having eigenvalues 1 and -1, with corresponding eigenvectors \( \left[ \begin{array}{c} \frac{\sqrt{3}}{2} \\ \frac{1}{2} \end{array} \right] \) and \( \left[ \begin{array}{c} \frac{-1}{2} \\ \frac{\sqrt{3}}{2} \end{array} \right] \), respectively. If \( A^{2021} = \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \), then \( a + b + c + d \) equals _________ (round off to 2 decimal places).

Let \( A \) be a \( 3 \times 3 \) real matrix such that \( I_3 + A \) is invertible and let \[ B = (I_3 + A)^{-1}(I_3 - A), \] where \( I_3 \) denotes the \( 3 \times 3 \) identity matrix. Then which one of the following statements is true?

Let \( M \) be the collection of all \( 3 \times 3 \) real symmetric positive definite matrices. Consider the set \[ S = \left\{ A \in M : A^{50} - \frac{1}{4} A^{48} = 0 \right\}, \] where \( 0 \) denotes the \( 3 \times 3 \) zero matrix. Then the number of elements in \( S \) equals