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Recent questions tagged linear-algebra

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Consider an $n \times n$ matrix $\text{A}$ and a non-zero $n \times 1$ vector $\text{p}$. Their product $Ap=\alpha ^{2}p$, where $\alpha \in \Re$ and $\alpha \notin \left \{ -1,0,1 \right \}$. Based on the given information, the eigen value of $A^{2}$ is: $\alpha$ $\alpha ^{2}$ $\surd{\alpha }$ $\alpha ^{4}$
asked Mar 1 in Linear Algebra jothee 4.9k points
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Let the superscript $\text{T}$ represent the transpose operation. Consider the function $f(x)=\frac{1}{2}x^TQx-r^Tx$, where $x$ and $r$ are $n \times 1$ vectors and $\text{Q}$ is a symmetric $n \times n$ matrix. The stationary point of $f(x)$ is $Q^{T}r$ $Q^{-1}r$ $\frac{r}{r^{T}r}$ $r$
asked Mar 1 in Linear Algebra jothee 4.9k points
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Consider a vector $\text{p}$ in $2$-dimensional space. Let its direction (counter-clockwise angle with the positive $\text{x}$-axis) be $\theta$. Let $\text{p}$ be an eigenvector of a $2\times2$ matrix $\text{A}$ with corresponding eigenvalue $\lambda ,\lambda > 0$. If we denote the ... ${p}'=\theta ,\left \| {p}' \right \|= \left \| p \right \|/\lambda$
asked Feb 22 in Linear Algebra gatecse 1.6k points
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A matrix $P$ is decomposed into its symmetric part $S$ and skew symmetric part $V$. If $S= \begin{pmatrix} -4 & 4 & 2 \\ 4 & 3 & 7/2 \\ 2 & 7/2 & 2 \end{pmatrix}, \: \: V= \begin{pmatrix} 0 & -2 & 3 \\ 2 & 0 & 7/2 \\ -3 & -7/2 & 0 \end{pmatrix}$ then matrix $P$ ... $\begin{pmatrix} -2 & 9/2 & -1 \\ -1 & 81/4 & 11 \\ -2 & 45/2 & 73/4 \end{pmatrix}$
asked Sep 18, 2020 in Linear Algebra jothee 4.9k points
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Let $\textbf{I}$ be a $100$ dimensional identity matrix and $\textbf{E}$ be the set of its distinct (no value appears more than once in $\textbf{E})$ real eigen values. The number of elements in $\textbf{E}$ is _________
asked Sep 18, 2020 in Linear Algebra jothee 4.9k points
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Multiplication of real valued square matrices of same dimension is associative commutative always positive definite not always possible to compute
asked Feb 19, 2020 in Linear Algebra jothee 4.9k points
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In matrix equation $[A] \{X\}=\{R\}$, $[A] = \begin{bmatrix} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{bmatrix} \{X\} = \begin{Bmatrix} 2 \\ 1 \\ 4 \end{Bmatrix} \text{ and} \{ R \} = \begin{Bmatrix} 32 \\ 16 \\ 64 \end{Bmatrix}$ One of the eigen values of matrix $[A]$ is $4$ $8$ $15$ $16$
asked Feb 9, 2019 in Linear Algebra Arjun 24.6k points
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The transformation matrix for mirroring a point in $x – y$ plane about the line $y=x$ is given by $\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} \\$ $\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} \\$ $\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \\$ $\begin{bmatrix} 0 & -1 \\ -1 & 0 \end{bmatrix}$
asked Feb 9, 2019 in Linear Algebra Arjun 24.6k points
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Consider the matrix $P=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ The number of distinct eigenvalues $0$ $1$ $2$ $3$
asked Feb 9, 2019 in Linear Algebra Arjun 24.6k points
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The set of equations $\begin{array}{l} x+y+z=1 \\ ax-ay+3z=5 \\ 5x-3y+az=6 \end{array}$ has infinite solutions, if $a=$ $-3$ $3$ $4$ $-4$
asked Feb 9, 2019 in Linear Algebra Arjun 24.6k points
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$x+2y+z=4$ $2x+y+2z=5$ $x-y+z=1$ The system of algebraic equations given above has a unique solution of $x=1$, $y=1$ and $z=1$ only the two solutions of $(x=1, y=1, z=1)$ and $(x=2, y=1, z=0)$ infinite number of solutions no feasible solution
asked Mar 20, 2018 in Linear Algebra Andrijana3306 1.5k points
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For the matrix $A = \begin{bmatrix} 5 & 3 \\ 1 & 3 \end{bmatrix}$, ONE of the normalized eigen vectors is given as $\begin{pmatrix} \dfrac{1}{2} \\ \dfrac{\sqrt{3}}{2} \end{pmatrix} \\$ $\begin{pmatrix} \dfrac{1}{\sqrt{2}} \\ \dfrac{-1}{\sqrt{2}} \end{pmatrix} \\$ ... $\begin{pmatrix} \dfrac{1}{\sqrt{5}} \\ \dfrac{2}{\sqrt{5}} \end{pmatrix}$
asked Mar 20, 2018 in Linear Algebra Andrijana3306 1.5k points
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If $A=\begin{bmatrix}1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 1 \end{bmatrix}$ then $\text{det}(A^{-1})$ is _______ (correct to two decimal palces).
asked Feb 17, 2018 in Linear Algebra Arjun 24.6k points
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The rank of the matrix $\begin{bmatrix} -4 & 1 & -1 \\ -1 & -1 & -1 \\ 7 & -3 & 1 \end{bmatrix}$ is $1$ $2$ $3$ $4$
asked Feb 17, 2018 in Linear Algebra Arjun 24.6k points
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Consider the matrix $A=\begin{bmatrix} 50 &70 \\ 70 & 80 \end{bmatrix}$ whose eigenvectors corresponding to eigenvalues $\lambda _{1}$ and $\lambda _{2}$ are $x_{1}=\begin{bmatrix} 70 \\ \lambda_{1}-50 \end{bmatrix}$ and $x_{2}=\begin{bmatrix} \lambda _{2}-80\\ 70 \end{bmatrix}$, respectively. The value of $x^{T}_{1} x_{2}$ is _________.
asked Feb 27, 2017 in Linear Algebra Arjun 24.6k points
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The determinant of a $2 \times 2$ matrix is $50$. If one eigenvalue of the matrix is $10$, the other eigenvalue is _________.
asked Feb 27, 2017 in Linear Algebra Arjun 24.6k points
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Consider the matrix $P=\begin{bmatrix} \dfrac{1}{\sqrt{2}} & 0 &\dfrac{1}{\sqrt{2}} \\ 0 & 1 & 0\\ -\dfrac{1}{\sqrt{2}} &0 & \dfrac{1}{\sqrt{2}} \end{bmatrix}$ Which one of the following statements about $P$ is INCORRECT ? Determinant of P is equal to $1$. $P$ is orthogonal. Inverse of $P$ is equal to its transpose. All eigenvalues of $P$ are real numbers.
asked Feb 27, 2017 in Linear Algebra Arjun 24.6k points
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The product of eigenvalues of the matrix $P$ is $P=\begin{bmatrix} 2 & 0 & 1\\ 4& -3 &3 \\ 0 & 2 & -1 \end{bmatrix}$ $-6$ $2$ $6$ $-2$
asked Feb 27, 2017 in Linear Algebra Arjun 24.6k points
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The number of linearly independent eigenvectors of matrix $A=\begin{bmatrix} 2 & 1 & 0\\ 0 &2 &0 \\ 0 & 0 & 3 \end{bmatrix}$ is _________
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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A real square matrix $\textbf{A}$ is called skew-symmetric if $A^T=A$ $A^T=A^{-1}$ $A^T=-A$ $A^T=A+A^{-1}$
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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The condition for which the eigenvalues of the matrix $A=\begin{bmatrix} 2 & 1\\ 1 & k \end{bmatrix}$ are positive, is $k > 1/2$ $k > −2$ $k > 0$ $k < −1/2$
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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The solution to the system of equations $\begin{bmatrix} 2 & 5\\-4 &3 \end{bmatrix}\begin{bmatrix} x\\y \end{bmatrix}=\begin{bmatrix} 2\\ -30 \end{bmatrix}$ is $6,2$ $-6,2$ $-6,-2$ $6,-2$
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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For a given matrix $P=\begin{bmatrix} 4+3i & -i\\ i & 4-3i \end{bmatrix}$, where $i=\sqrt{-1}$, the inverse of matrix $P$ is $P=\displaystyle{\frac{1}{24}}\begin{bmatrix} 4-3i & i\\ -i & 4+3i \end{bmatrix} \\$ ... $P=\displaystyle{\frac{1}{25}}\begin{bmatrix} 4+3i & -i\\ i & 4-3i \end{bmatrix} \\$
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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The lowest eigenvalue of the $2\times 2$ matrix $\begin{bmatrix} 4 & 2\\ 1 & 3 \end{bmatrix}$ is ________
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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At least one eigenvalue of a singular matrix is positive zero negative imaginary
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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If any two columns of a determinant $P=\begin{bmatrix} 4 & 7 & 8\\ 3 & 1 & 5\\ 9 & 6 & 2 \end{bmatrix}$ are interchanged, which one of the following statements regarding the value of the determinant is CORRECT? Absolute value remains unchanged ... . Both absolute value and sign will change. Absolute value will change but sign will not change. Both absolute value and sign will remain unchanged.
asked Feb 24, 2017 in Linear Algebra Arjun 24.6k points
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Which one of the following equations is a correct identity for arbitrary $3 \times 3$ real matrices $P$, $Q$ and $R$? $P(Q+R)=PQ+RP$ $(P-Q)^2 = P^2 -2PQ -Q^2$ $\text{det } (P+Q)= \text{det } P+ \text{det } Q$ $(P+Q)^2=P^2+PQ+QP+Q^2$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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Consider a $3×3$ real symmetric matrix S such that two of its eigenvalues are $a\neq 0$, $b\neq 0$ with respective eigenvectors $\begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$, $\begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix}$ If $a\neq b$ then $x_1y_1+x_2y_2+x_3y_3$ equals $a$ $b$ $ab$ $0$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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If there are $m$ sources and $n$ destinations in a transportation matrix, the total number of basic variables in a basic feasible solution is $m + n$ $m + n + 1$ $m + n − 1$ $m$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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One of the eigen vectors of the matrix $\begin{bmatrix} -5 & 2\\ -9 & 6 \end{bmatrix}$ is $\begin{Bmatrix} -1\\ 1 \end{Bmatrix} \\$ $\begin{Bmatrix} -2\\ 9 \end{Bmatrix} \\$ $\begin{Bmatrix} 2\\ -1 \end{Bmatrix} \\$ $\begin{Bmatrix} 1\\ 1 \end{Bmatrix} \\$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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The matrix form of the linear syatem $\dfrac{dx}{dt}=3x-5y$ and $\dfrac{dy}{dt}=4x+8y$ is $\dfrac{d}{dt}\begin{Bmatrix} x\\y \end{Bmatrix}=\begin{bmatrix} 3 & -5\\ 4& 8 \end{bmatrix}\begin{Bmatrix} x\\y \end{Bmatrix} \\$ ... $\dfrac{d}{dt}\begin{Bmatrix} x\\y \end{Bmatrix}=\begin{bmatrix} 4 & 8\\ 3& -5 \end{bmatrix}\begin{Bmatrix} x\\y \end{Bmatrix}$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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Given that the determinant of the matrix $\begin{bmatrix} 1 & 3 & 0\\ 2 & 6 & 4\\ -1 & 0 & 2 \end{bmatrix}$ is $-12$, the determinant of the matrix $\begin{bmatrix} 2 & 6 & 0\\ 4 & 12 & 18\\ -2 & 0 & 4 \end{bmatrix}$ is $-96$ $-24$ $24$ $96$
asked Feb 19, 2017 in Linear Algebra Arjun 24.6k points
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The eigen values of a symmetric matrix are all complex with non-zero positive imaginary part. complex with non-zero negative imaginary part. real. pure imaginary.
asked Feb 19, 2017 in Linear Algebra piyag476 1.4k points
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