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Recent questions tagged linear-algebra

47 votes
14 answers
331
GATE CSE 2021 Set 2 | Question: 24
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T$. The rank of $P$ is __________
Suppose that $P$ is a $4 \times 5$ matrix such that every solution of the equation $\text{Px=0}$ is a scalar multiple of $\begin{bmatrix} 2 & 5 & 4 &3 & 1 \end{bmatrix}^T...
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18 votes
5 answers
332
GATE CSE 2021 Set 1 | Question: 52
Consider the following matrix.$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$The largest eigenvalue of the above matrix is __________.
Consider the following matrix.$$\begin{pmatrix} 0 & 1 & 1 & 1\\ 1& 0& 1 & 1\\ 1& 1 & 0 & 1 \\1 & 1 & 1 & 0 \end{pmatrix}$$The largest eigenvalue of the above matrix is __...
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5 votes
1 answer
333
GATE Overflow Test Series | Mock GATE | Test 5 | Question: 24
Let $P$ be a matrix of order $3 \times 3$ whose elements are real numbers. If $P^2 = 0$, then the eigenvalues of $P$ are ________ $0,0,0$ $0,0,1$ $0,1,1$ $1,1,1$
Let $P$ be a matrix of order $3 \times 3$ whose elements are real numbers. If $P^2 = 0$, then the eigenvalues of $P$ are ________$0,0,0$$0,0,1$$0,1,1$$1,1,1$
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2 votes
1 answer
336
CMI-2018-DataScience-A: 1
If $P$ is an invertible matrix and $A=PBP^{-1},$ then which of the following statements are necessarily true? $B=P^{-1}AP$ $|A|=|B|$ $A$ is invertible if and only if $B$ is invertible $B^T=A^T$
If $P$ is an invertible matrix and $A=PBP^{-1},$ then which of the following statements are necessarily true?$B=P^{-1}AP$$|A|=|B|$$A$ is invertible if and only if $B$ is ...
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0 votes
3 answers
337
CMI-2018-DataScience-A: 2
Let ... $|A|=|B|$ $|C|=|D|$ $|B|=-|C|$ $|A|=-|D|$
Let $A=\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{bmatrix}, C=\begin{bmatrix} 4 & 5 & 6...
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2 votes
2 answers
338
CMI-2018-DataScience-A: 3
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true? $x^Tx$ is a $3\times 3$ matrix $xx^T$ is a $3\times 3$ matrix $xx^T$ is a $1\times 1$ matrix $xx^T=x^Tx$
Let $x=\begin{bmatrix} 3& 1 & 2 \end{bmatrix}$. Which of the following statements are true?$x^Tx$ is a $3\times 3$ matrix$xx^T$ is a $3\times 3$ matrix$xx^T$ is a $1\time...
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1 votes
2 answers
342
CMI-2020-DataScience-B: 2
Consider the matrix $A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Find $A^n,$ in terms of $n,$ for $n\geq2.$
Consider the matrix $A=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$. Find $A^n,$ in terms of $n,$ for $n\geq2.$
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4 votes
2 answers
343
GATE Overflow Test Series | Mock GATE | Test 3 | Question: 22
Let $P$ be an invertible matrix of order $3 \times 3$ and $\mid P \mid = 5$, then the value of $\mid \text{adj}(\text{adj}(P))\mid$ is ________
Let $P$ be an invertible matrix of order $3 \times 3$ and $\mid P \mid = 5$, then the value of $\mid \text{adj}(\text{adj}(P))\mid$ is ________
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1 votes
3 answers
347
NIELIT 2016 MAR Scientist C - Section B: 8
The eigenvalues of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$ are $\text{5 and -5}$ $\text{5 and -1}$ $\text{1 and -5}$ $\text{2 and 3}$
The eigenvalues of the matrix $\begin{bmatrix}1 & 2\\ 4 & 3 \end{bmatrix}$ are$\text{5 and -5}$$\text{5 and -1}$$\text{1 and -5}$$\text{2 and 3}$
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0 votes
1 answer
350
NIELIT 2017 OCT Scientific Assistant A (IT) - Section B: 7
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The value of $’n’$ is $2$ $3$ $5$ $4$
$M$ is a square matrix of order $’n’$ and its determinant value is $5.$ If all the elements of $M$ are multiplied by $2,$ its determinant value becomes $40.$ The valu...
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2 votes
1 answer
352
NIELIT 2016 MAR Scientist B - Section B: 4
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$ $-76$ $-28$ $+28$ $+72$
What is the determinant of the matrix $\begin{bmatrix}5&3&2\\1&2&6\\3&5&10\end{bmatrix}$$-76$$-28$$+28$$+72$
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4 votes
1 answer
353
NIELIT 2016 MAR Scientist B - Section B: 6
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.
The system of simultaneous equations$x+2y+z=6\\2x+y+2z=6\\x+y+z=5$hasunique solution.infinite number of solutions.no solution.exactly two solutions.
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3 votes
2 answers
354
NIELIT 2016 MAR Scientist B - Section B: 12
If $A$ and $B$ are square matrices of size $n\times n$, then which of the following statements is not true? $\det(AB)=\det(A) \det(B)$ $\det(kA)=k^n \det(A)$ $\det(A+B)=\det(A)+\det(B)$ $\det(A^T)=1/\det(A^{-1})$
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...
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3 votes
3 answers
355
NIELIT 2016 DEC Scientist B (IT) - Section B: 26
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+​ \sqrt-1)$ and $3$. The determinant of $P$ is ________. $0$ $1$ $15$ $-1$
Two eigenvalues of a $3\times3$ real matrix $P$ are $(2+​ \sqrt-1)$ and $3$. The determinant of $P$ is ________.$0$$1$$15$$-1$
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2 votes
2 answers
356
NIELIT 2016 DEC Scientist B (CS) - Section B: 21
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is: $D^{-1}C^{-1}A^{-1}$ $CDA$ $ADC$ Does not necessarily exist.
Let $A,B,C,D$ be $n\times n$ matrices, each with non-zero determinant. If $ABCD=1$, then $B^{-1}$ is:$D^{-1}C^{-1}A^{-1}$$CDA$$ADC$Does not necessarily exist.
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1 votes
1 answer
357
NIELIT 2017 DEC Scientist B - Section B: 60
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 $M_1$
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 t...
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