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$$\small{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline \textbf{Year}&\textbf{2019}&\textbf{2018}&\textbf{2017-1}&\textbf{2017-2}&\textbf{2016-1}&\textbf{2016-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum} \\\hline\textbf{1 Mark Count}&1&1&1&1&1&2&1&1.2&2 \\\hline\textbf{2 Marks Count}&1&1&2&1&0&0&0&0.8&2 \\\hline\textbf{Total Marks}&3&3&5&3&1&2&\bf{1}&\bf{2.5}&\bf{5}\\\hline \end{array}}}$$

# Recent questions in Linear Algebra

1
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$ $-76$ $-28$ $+28$ $+72$
2
The system of simultaneous equations $x+2y+z=6\\2x+y+2z=6\\x+y+z=5$ has unique solution. infinite number of solutions. no solution. exactly two solutions.
3
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
4
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+​ \sqrt-1)$ and $3$. The determinant of $P$ is ________. $0$ $1$ $15$ $-1$
1 vote
5
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
6
Consider two matrices $M_1$ and $M_2$ with $M_1^*M_2=0$ and $M_1$ is non singular. Then which of the following is true? $M_2$ is non singular $M_2$ is null matrix $M_2$ is the identity matrix $M_2$ is transpose of $M_1$
1 vote
7
$AVA=A$ is called : Identity law De Morgan’s law Idempotent law Complement law
8
How many corner does a cube have in 4 dimensions? How many 3D faces? Now by observation we can tell that, an n-dimensional cube has $2^n$ corners. 1D cube which is a line have $2^1$ corners 2D cube which is a square have $2^2$ corners 3D cube have $2^3$ corners ... 8 three-dimension cubes. but this is the question i'm not able to answer. How every N-cube have $|2n|$ cubes of dimension (N-1)?
9
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
10
Let $A$ and $B$ be two $n \times n$ matrices over real numbers. Let rank($M$) and $\text{det}(M)$ denote the rank and determinant of a matrix $M$, respectively. Consider the following statements. $\text{rank}(AB) = \text{rank }(A) \text{rank }(B)$ ... Which of the above statements are TRUE? I and II only I and IV only II and III only III and IV only
11
The hour needle of a clock is malfunctioning and travels in the anti-clockwise direction, i.e., opposite to the usual direction, at the same speed it would have if it was working correctly. The minute needle is working correctly. Suppose the two needles show the correct time at $12$ noon, thus ... ? $\dfrac{10}{11}$ hour $\dfrac{11}{12}$ hour $\dfrac{12}{13}$ hour $\dfrac{19}{22}$ hour One hour
12
Let $A$ be am $n\times n$ invertible matrix with real entries whose column sums are all equal to $1$. Consider the following statements: Every column in the matrix $A^{2}$ sums to $2$ Every column in the matrix $A^{3}$ sums to $3$ Every column in the matrix $A^{-1}$ ... $(3)$ is correct but not statements $(1)$ or $(2)$ all the $3$ statements $(1),(2),$ and $(3)$ are correct
13
Let $d\geq 4$ and fix $w\in \mathbb{R}.$ Let $S = \{a = (a_{0},a_{1},\dots ,a_{d})\in \mathbb{R}^{d+1}\mid f_{a}(w) = 0\: \text{and}\: f'_{a}(w) = 0\},$ where the polynomial function $f_{a}(x)$ ... $d$-dimensional vector subspace of $\mathbb{R}^{d+1}$ $S$ is a $(d-1)$-dimensional vector subspace of $\mathbb{R}^{d+1}$ None of the other options
1 vote
14
Let $M$ be a real $n\times n$ matrix such that for$every$ non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x> 0.$ Then Such an $M$ cannot exist Such $Ms$ exist and their rank is always $n$ Such $Ms$ exist, but their eigenvalues are always real No eigenvalue of any such $M$ can be real None of the above
1 vote
15
If $M$ is a $3 \times 3$ matrix such that $\begin{bmatrix} 0 & 1 & 2 \end{bmatrix}M=\begin{bmatrix}1 & 0 & 0 \end{bmatrix}$ and $\begin{bmatrix}3 & 4 & 5 \end{bmatrix} M = \begin{bmatrix}0 & 1 & 0 \end{bmatrix}$ then $\begin{bmatrix}6 & 7 & 8 \end{bmatrix}M$ is ... $\begin{bmatrix}0 & 0 & 1 \end{bmatrix}$ $\begin{bmatrix} -1 & 2 & 0 \end{bmatrix}$ $\begin{bmatrix} 9 & 10 & 8 \end{bmatrix}$
16
The values of $\eta$ for which the following system of equations $\begin{array} {} x & + & y & + & z & = & 1 \\ x & + & 2y & + & 4z & = & \eta \\ x & + & 4y & + & 10z & = & \eta ^2 \end{array}$ has a solution are $\eta=1, -2$ $\eta=-1, -2$ $\eta=3, -3$ $\eta=1, 2$
1 vote
17
The determinant $\begin{vmatrix} b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y \end{vmatrix}$ equals $\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ $2\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ $3\begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix}$ None of these
1 vote
Suppose that $A$ is a $3 \times 3$ real matrix such that for each $u=(u_1, u_2, u_3)’ \in \mathbb{R}^3, \: u’Au=0$ where $u’$ stands for the transpose of $u$. Then which one of the following is true? $A’=-A$ $A’=A$ $AA’=I$ None of these
The value of $\lambda$ such that the system of equation $\begin{array}{} 2x & – & y & + & 2z & = & 2 \\ x & – & 2y & + & z & = & -4 \\ x & + & y & + & \lambda z & = & 4 \end{array}$ has no solution is $3$ $1$ $0$ $-3$
For the matrices $A = \begin{pmatrix} a & a \\ 0 & a \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$, $(B^{-1}AB)^3$ is equal to $\begin{pmatrix} a^3 & a^3 \\ 0 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 3a^3 \\ 0 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ 3a^3 & a^3 \end{pmatrix}$ $\begin{pmatrix} a^3 & 0 \\ -3a^3 & a^3 \end{pmatrix}$