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Questions without an upvoted answer in Linear Algebra

1 1 vote
1 1 answer
GATE IN 2026 | Question: 1
A matrix $P \in \mathbb{R}^{3 \times 3}$ satisfies $P^{3}-4 P^{2}+5 P-2 I=0$ where $I$ is a $3 \times 3$ identity matrix.The set of all possible eigenvalues of matrix $P$...
4 4 votes
2 2 answers
GATE IN 2025 | Question: 1
A $2 n \times 2 n$ matrix $A=\left[a_{i j}\right]$ has its elements as$$ a_{i j}=\begin{cases} \beta & \text { if }(i+j) \text { is odd } \\ -\beta & \text { if }(i+j) \t...
0 0 votes
0 0 answers
GATE IN 2025 | Question: 22
If one of the eigenvectors of the matrix $A=\left[\begin{array}{cc}-1 & -1 \\ x & -4\end{array}\right]$ is along the direction of $\left[\begin{array}{c}\alpha \\ 2 \alph...
2 2 votes
1 1 answer
GATE IN 2024 | Question: 9
A matrix $M$ is constructed by stacking three column vectors $v_{1}, v_{2}, v_{3}$ as$$ M=\left[\begin{array}{lll} v_{1} & v_{2} & v_{3} \end{array}\right]$$Choose the se...
0 0 votes
1 1 answer
GATE IN 2024 | Question: 16
A $3 \times 3$ matrix $P$ with all real elements has eigenvalues $\dfrac{1}{4}, 1$, and $-2$. The value of $\mid P^{-1}\mid$ is $\_\_\_\_\_\_$ (rounded off to nearest int...
0 0 votes
0 0 answers
GATE IN 2023 | Question: 1
Choose solution set $S$ corresponding to the systems of two equations$$\begin{array}{r}x-2 y+z=0 \\x-z=0\end{array}$$Note: $\mathscr{R}$ denotes the set of real numbers$S...
1 1 vote
0 0 answers
GATE IN 2022 | Question: 14
Given $\text{M} = \begin{bmatrix} 2 & 3 & 7 \\ 6 & 4 & 7 \\ 4 & 6 & 14 \end{bmatrix},$ Which of the following statement(s) is/are correct?The rank of $\text{M}$ is $2$The...
0 0 votes
0 0 answers
GATE IN 2022 | Question: 39
The matrix $\text{A} = \begin{bmatrix} 4 & 3 \\ 9 & -2 \end{bmatrix}$ has eigenvalues $-5$ and $7.$The eigenvectors(s) is/are __________ $\begin{bmatrix} 1\\ 1 \end{bmat...
1 1 vote
0 0 answers
GATE IN 2021 | Question: 1
Consider the row vectors $v=(1,0)$ and $w=(2,0)$. The rank of the matrix $M=2v^{T}v+3w^{T}w$, where the superscript $\text{T}$ denotes the transpose, is$1$$2$$3$$4$
1 1 vote
0 0 answers
GATE IN 2021 | Question: 25
The determinant of the matrix $\text{M}$ shown below is _______________. $$M=\begin{bmatrix} 1 & 2 & 0 & 0\\ 3 & 4 & 0 & 0\\ 0 & 0 & 4 & 3\\ 0 & 0 & 2 & 1 \end{bmatrix}$$
0 0 votes
0 0 answers
GATE IN 2021 | Question: 38
Given $A=\begin{pmatrix} 2 & 5\\ 0 & 3 \end{pmatrix}$. The value of the determinant $\left | A^{4}-5A^{3}+6A^{2}+2I \right |=$ _______________.
0 0 votes
0 0 answers
GATE2020 IN: 26
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}$$\begin {bmatrix} 1...
0 0 votes
0 0 answers
GATE2020: 3
A set of linear equations is given in the form $Ax=b$, where A is a $2\times 4$ matrix with real number entries and $b\neq 0.$ will it be possible to solve for $x$ and ob...
0 0 votes
0 0 answers
GATE IN 2017 | Question: 1
If $\text{v}$ is a non-zero vector of dimensions $3\times1$, then the matrix $A=VV^T$ has a rank = ____________.
1 1 vote
0 0 answers
GATE IN 2017 | Question: 2
The figure shows a shape $\text ABC$ and its mirror image $\text A_1B_1C_1$ across the horizontal axis $\text (X-axis)$. The coordinate transformation matrix that maps $\...
0 0 votes
0 0 answers
GATE IN 2017 | Question: 4
The eigenvalues of the matrix $A=\begin{bmatrix}1 &-1 &5\\0 &5 &6\\0 &-6 &5\end{bmatrix}$ are$-1,\;5,\;6$$1,\;-5\pm j6$$1,\;5\pm j6$$1,\;5,\;5$
0 0 votes
0 0 answers
GATE2019 IN: 16
A 3 x 3 matrix has eigenvalues 1, 2 and 5. The determinant of the matrix is $\_\_\_\_$.
0 0 votes
0 0 answers
GATE2016-28
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 $\_\_\_\_\_...
0 0 votes
0 0 answers
GATE2015-11
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$$n+r$
0 0 votes
0 0 answers
GATE2014-26
A scalar valued function is defined as $f(X)=X^TAX+b^TX+c,$ where A is a symmetric positive definite matrix with dimension $n\times n$; b and X are vectors of dimension $...
0 0 votes
0 0 answers
GATE2014-28
For the matrix $A$ satisfying the equation given below, the eigenvalues are$$[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\en...
0 0 votes
0 0 answers
GATE IN 2013 | Question: 27
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}...
0 0 votes
0 0 answers
GATE IN 2013 | Question: 1
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$$3$
0 0 votes
0 0 answers
GATE IN 2012 | Question: 27
Given that$A=\begin{bmatrix}-5 &-3\\2 &0\end{bmatrix}$ and $I=\begin{bmatrix}1 & 0\\0 &1\end{bmatrix}$, the value of $A^3$ is$15A+12I$$19A+30I$$17A+15I$$17A+21I$
0 0 votes
0 0 answers
GATE2018IN: 28
Consider the following system of linear equations: 3x + 2ky = -2 kx + 6y =2Here x and y are the unknows and k is real c...
0 0 votes
0 0 answers
GATE2018IN: 1
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, 00,0,10,1,11,1,1
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