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Recent questions tagged linear-algebra

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41
GATE ECE 2015 Set 2 | Question: 2
The value of $x$ for which all the eigen-values of the matrix given below are real is $\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$ $5+j$ $5-j$ $1-5j$ $1+5j$
The value of $x$ for which all the eigen-values of the matrix given below are real is $$\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$$$5+j$$5-j$$1-5j$$1...
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42
GATE ECE 2015 Set 2 | Question: 46
The state variable representation of a system is given as $\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$ $y=\begin{bmatrix} 0 &1 \end{bmatrix} x$ The response $y(t)$ is $\sin(t)$ $1-e^{t}$ $1-\cos(t)$ $0$
The state variable representation of a system is given as$\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$$y=\begin{bm...
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43
GATE ECE 2015 Set 1 | Question: 1
Consider a system of linear equations: $x-2y+3z=-1, \\ x-3y+4z=1, \text{ and } \\ -2x+4y-6z=k.$ The value of $k$ for which the system has infinitely many solutions is ___________
Consider a system of linear equations:$$x-2y+3z=-1, \\ x-3y+4z=1, \text{ and } \\ -2x+4y-6z=k.$$ The value of $k$ for which the system has infinitely many solutions is __...
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44
GATE ECE 2015 Set 1 | Question: 5
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 \end{bmatrix}$ is _________.
The value of $p$ such that the vector $\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ is an eigenvector of the matrix $\begin{bmatrix} 4 & 1 & 2 \\ p & 2 & 1 \\ 14 & -4 & 10 ...
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45
GATE ECE 2015 Set 1 | Question: 43
Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ ... $\begin{bmatrix}p, & q, & r \end{bmatrix} = \begin{bmatrix} c, & b, & a \end{bmatrix}$
Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ are related as,$$\begin{bmatrix}A \\ B \\ C \end{bmatrix} = \be...
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49
GATE ECE 2014 Set 2 | Question: 1
The determinant of matrix $A$ is $5$ and the determinant of matrix B is $40$. The determinant of matrix $AB$ is ________
The determinant of matrix $A$ is $5$ and the determinant of matrix B is $40$. The determinant of matrix $AB$ is ________
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51
GATE ECE 2014 Set 2 | Question: 28
The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.
The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.
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52
GATE ECE 2014 Set 1 | Question: 1
For matrices of same dimension $M, N$ and scalar $c$, which one of these properties DOES NOT ALWAYS hold? $(M^{T})^{T} = M$ $(cM)^{T} = c(M)^{T}$ $(M+N)^{T} = M^{T} + N^{T}$ $MN = NM$
For matrices of same dimension $M, N$ and scalar $c$, which one of these properties DOES NOT ALWAYS hold?$(M^{T})^{T} = M$$(cM)^{T} = c(M)^{T}$$(M+N)^{T} = M^{T} + N^{T}...
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53
GATE ECE 2014 Set 1 | Question: 4
A real $(4 \times 4)$ matrix $A$ satisfies the equation $A^{2} = I$, where $I$ is the $(4 \times 4)$ identity matrix. The positive eigen value of $A$ is ______.
A real $(4 \times 4)$ matrix $A$ satisfies the equation $A^{2} = I$, where $I$ is the $(4 \times 4)$ identity matrix. The positive eigen value of $A$ is ______.
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54
GATE ECE 2014 Set 1 | Question: 29
Consider the matrix ... $\alpha$ is a non-negative real number. The value of $\alpha$ for which $\text{det(P)} = 0$ is _______.
Consider the matrix $$J_{6} = \begin{bmatrix} 0&0 &0 &0 &0 &1 \\ 0& 0& 0& 0& 1&0 \\ 0& 0& 0& 1& 0&0 \\ 0&0 & 1& 0&0 &0 \\0 &1 &0 &0 &0 &0 \\1 &0 &0 &0 & 0& 0\end{bmatrix}...
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56
GATE ECE 2013 | Question: 19
The minimum eigenvalue of the following matrix is $\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$ $0$ $1$ $2$ $3$
The minimum eigenvalue of the following matrix is$$\begin{bmatrix} 3& 5& 2\\5 &12 &7 \\2 &7 & 5\end{bmatrix}$$$0$$1$$2$$3$
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57
GATE ECE 2012 | Question: 47
Given that $A=\begin{bmatrix} -5 &-3 \\ 2 &0\end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 \\ 0 &1\end{bmatrix}$, the value of $A^3$ is $15\:A+12\:I$ $19\:A+30\:I$ $17\:A+15\:I$ $17\:A+21\:I$
Given that$A=\begin{bmatrix} -5 &-3 \\ 2 &0\end{bmatrix}$ and $I=\begin{bmatrix} 1 & 0 \\ 0 &1\end{bmatrix}$, the value of $A^3$ is$15\:A+12\:I$$19\:A+30\:I$$17\:A+15\:I$...
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58
GATE ECE 2018 | Question: 22
Consider matrix $A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix}$ and vector $x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}.$ The number of distinct real value of $k$ for which the equation $Ax=0$ has infinitely many solutions is _________.
Consider matrix $A=\begin{bmatrix} k & 2k\\ k^{2}-k & k^{2} \end{bmatrix}$ and vector $x=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}.$ The number of distinct real value o...
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60
GATE ECE 2017 Set 2 | Question: 1
The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0& 0& 1& -1& \end{bmatrix}$ is ________.
The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0&...
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61
GATE ECE 2017 Set 1 | Question: 2
The rank of the matrix $\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$ is $0$ $1$ $2$ $3$
The rank of the matrix $$\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$$ is$0$$1$$2$$3$
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62
GATE ECE 2017 Set 1 | Question: 1
Consider the 5 $\times$ 5 matrix $\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$ It is given that A has only one real eigenvalue. Then the real eigenvalue of A is $-2.5$ $0$ $15$ $25$
Consider the 5 $\times$ 5 matrix$$\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$$It is given that A has only one real eigenv...
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