List of top Matrix Theory Questions on Matrix Algebra asked in GATE ST

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