List of top Statistics Questions on Linear Algebra asked in GATE ST

Consider the following two subspaces of \(\mathbb{R}^4\) \[ W_1=\{(x_1,x_2,x_3,x_4):x_1+x_2+x_3+x_4=0\} \] \[ W_2=\{(x_1,x_2,x_3,x_4):x_1+2x_2+3x_3+4x_4=0\}. \] Which of the following statements is correct?

Consider the system of linear equations
\[ x+y+z=1, \]\[ 2x+y+3z=6, \]\[ 3x+2y+kz=k+1. \]If the above system has no solution, then the value of \(k\) equals _______ (answer in integer).

Consider the subspace
\[ U = \{(x_1, x_2, x_3, x_4) \in \mathbb{R}^4 : x_1+x_2+x_4=0,\ x_3=0\}. \]
Let \(U^{\perp}\) be its orthogonal complement. The value of \(\dim(U^{\perp})\) equals ________ (answer in integer).

Let \(A\in M_n(\mathbb{R})\) be an \(n\times n\) real matrix, \(n\geq2\). Consider the following two statements:
(I) If \(\lambda\in\mathbb{C}\) is an eigenvalue of \(A\), then its complex conjugate \(\bar\lambda\) is also an eigenvalue.
(II) If \(v=(v_1,v_2,\ldots,v_n)\in\mathbb{C}^n\) is an eigenvector corresponding to eigenvalue \(\lambda=x+iy\), \(y\neq0\), then
\[ \text{Re}(v)=(\text{Re}(v_1),\ldots,\text{Re}(v_n))\quad\text{and}\quad\text{Im}(v)=(\text{Im}(v_1),\ldots,\text{Im}(v_n)) \]
are linearly independent vectors over \(\mathbb{R}\).

Which of the following statements is correct?

Consider the system of equations
\[ x+2y-z=a \]
\[ x+y+3z=b \]
\[ 2x+3y+2z=c \]

Consider the following statements:
(I) For every \((a,b,c)\in\mathbb{R}^3\), the above system has a solution.
(II) For \((a,b,c)=(0,0,0)\), the solution set is given by
\[ \{(-7t,\ 4t,\ t):t\in\mathbb{R}\}. \]
Which of the following statements is correct?

Consider the matrix
\[ A=\begin{pmatrix} \lambda_1 & 1 & 0 \\ 0 & \lambda_2 & 1 \\ 0 & 0 & \lambda_3 \end{pmatrix}, \qquad \lambda_1,\lambda_2,\lambda_3\in\mathbb{R}. \]
Which of the following statements is/are correct?

Consider the real symmetric matrix \(A=(a_{ij})\) given by \[ A=\begin{pmatrix}3 & 1 & 1\\ 1 & 0 & 2\\ 1 & 2 & 0\end{pmatrix}. \] Consider the set \[ S=\left\{x=(x_1,x_2,x_3)\in\mathbb{R}^3 : \sum_{j=1}^{3}\sum_{i=1}^{3}a_{ij}x_ix_j=1\right\}. \] Which of the following statements is/are correct?