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Recent questions and answers in Linear Algebra

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Consider the row vectors $v=(1,0)$ and $w=(2,0)$. The rank of the matrix $M=2v^{T}v+3w^{T}w$, where the superscript $\text{T}$ denotes the transpose, is $1$ $2$ $3$ $4$
asked Feb 19 in Linear Algebra Arjun 2.7k points
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The determinant of the matrix $\text{M}$ shown below is _______________. $M=\begin{bmatrix} 1 & 2 & 0 & 0\\ 3 & 4 & 0 & 0\\ 0 & 0 & 4 & 3\\ 0 & 0 & 2 & 1 \end{bmatrix}$
asked Feb 19 in Linear Algebra Arjun 2.7k points
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Given $A=\begin{pmatrix} 2 & 5\\ 0 & 3 \end{pmatrix}$. The value of the determinant $\left | A^{4}-5A^{3}+6A^{2}+2I \right |=$ _______________.
asked Feb 19 in Linear Algebra Arjun 2.7k points
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Consider the matrix $M=\begin {bmatrix} 1&-1&0\\1&-2&1\\0&-1&1\end{bmatrix}$. One of the eigenvectors of $M$ is $\begin {bmatrix} 1\\-1\\1\end{bmatrix}$ $\begin {bmatrix} 1\\1\\-1\end{bmatrix}$ $\begin {bmatrix} -1\\1\\-1\end{bmatrix}$ $\begin {bmatrix} 1\\1\\1\end{bmatrix}$
asked Nov 3, 2020 in Linear Algebra soujanyareddy13 2.7k points
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A set of linear equations is given in the form $Ax=b$, where A is a $2\times 4$ matrix with real number entries and $b\neq 0.$ will it be possible to solve for $x$ and obtain a unique solution by multiplying both left and right sides of the equation ... a unique solution provided the matrix $A^T A$ is well conditioned Yes, can obtain a unique solution provided the matrix $A$ is well conditioned
asked Nov 3, 2020 in Linear Algebra soujanyareddy13 2.7k points
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If $\text{v}$ is a non-zero vector of dimensions $3\times1$, then the matrix $A=VV^T$ has a rank = ____________.
asked Nov 1, 2020 in Linear Algebra soujanyareddy13 2.7k points
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The figure shows a shape $\text ABC$ and its mirror image $\text A_1B_1C_1$ across the horizontal axis $\text (X-axis)$. The coordinate transformation matrix that maps $\text ABC$ to $\text A_1B_1C_1$ is $\begin{bmatrix}0&1\\1 &0\end{bmatrix}$ $\begin{bmatrix}0 &1\\-1 &0\end{bmatrix}$ $\begin{bmatrix}-1 &0\\0 &1\end{bmatrix}$ $\begin{bmatrix}1 &0\\0 &-1\end{bmatrix}$
asked Nov 1, 2020 in Linear Algebra soujanyareddy13 2.7k points
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The eigenvalues of the matrix $A=\begin{bmatrix}1 &-1 &5\\0 &5 &6\\0 &-6 &5\end{bmatrix}$ are $-1,\;5,\;6$ $1,\;-5\pm j6$ $1,\;5\pm j6$ $1,\;5,\;5$
asked Nov 1, 2020 in Linear Algebra soujanyareddy13 2.7k points
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A 3 x 3 matrix has eigenvalues 1, 2 and 5. The determinant of the matrix is $\_\_\_\_$.
asked Feb 10, 2019 in Linear Algebra Arjun 2.7k points
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Consider the matrix $A= \begin{pmatrix} 2 & 1 & 1\\ 2& 3& 4\\ -1& -1 & -2 \end{pmatrix} $ whose eigenvalues are $1, -1$ and $3$. Then Trace of $(A^3-3A^2)$ is $\_\_\_\_\_\_.$
asked Mar 27, 2018 in Linear Algebra Milicevic3306 7.9k points
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Let $A$ be an $n\times n$ matrix with rank $r(0<r<n).$ Then $\text{Ax=0}$ has $p$ independent solutions, where $p$ is $r$ $n$ $n-r$ $n+r$
asked Mar 27, 2018 in Linear Algebra Milicevic3306 7.9k points
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A scalar valued function is defined as $f(X)=X^TAX+b^TX+c,$ where A is a symmetric positive definite matrix with dimension $n\times n$; b and X are vectors of dimension $n\times 1$. The minimum value of f(x) will occur when X equals $(A^TA)^{-1}b$ $-(A^TA)^{-1}b$ $-(\frac{A^{-1}b}{2})$ $\frac{A^{-1}b}{2}$
asked Mar 26, 2018 in Linear Algebra Milicevic3306 7.9k points
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For the matrix $A$ satisfying the equation given below, the eigenvalues are $[A] \begin {bmatrix} 1&2&3\\7&8&9\\4&5&6\end{bmatrix}=\begin {bmatrix} 1&2&3\\4&5&6\\7&8&9\end{bmatrix}$ $(1,-j,j)$ $(1,1,0)$ $(1,1,-1)$ $(1,0,0)$
asked Mar 26, 2018 in Linear Algebra Milicevic3306 7.9k points
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One pair of eigenvectors corresponding to the two eigenvalues of the matrix $\begin{bmatrix}0&-1\\1&0\end{bmatrix}$ is $\begin{bmatrix}1\\-j\end{bmatrix}$,$\begin{bmatrix}j\\-1\end{bmatrix}$ $\begin{bmatrix}0\\1\end{bmatrix}$,$\begin{bmatrix}-1\\0\end{bmatrix}$ ... $\begin{bmatrix}0\\1\end{bmatrix}$ $\begin{bmatrix}1\\j\end{bmatrix}$,$\begin{bmatrix}j\\1\end{bmatrix}$
asked Mar 26, 2018 in Linear Algebra Milicevic3306 7.9k points
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The dimension of the null space of the matrix $\begin{bmatrix} 0&1&1\\1&-1&0\\-1&0&-1 \end{bmatrix}$ is $0$ $1$ $2$ $3$
asked Mar 26, 2018 in Linear Algebra Milicevic3306 7.9k points
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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 $15A+12I$ $19A+30I$ $17A+15I$ $17A+21I$
asked Mar 25, 2018 in Linear Algebra Milicevic3306 7.9k points
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Consider the following system of linear equations: 3x + 2ky = -2 kx + 6y =2 Here x and y are the unknows and k is real constant. The value of k for which there are infinite number of solutions is 3 1 -3 -6
asked Feb 20, 2018 in Linear Algebra gatecse 1.4k points
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Let N be a 3 by 3 matrix with real numbers entries. The matrix N is such that N$^2$ = 0. The eigen values of N are 0, 0, 0 0,0,1 0,1,1 1,1,1
asked Feb 20, 2018 in Linear Algebra gatecse 1.4k points
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