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Recent questions tagged goclasses2025_csda_wq4

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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 1
Consider the vectors $\mathbf{v}_1$ and $\mathbf{v}_2$ given by$$\mathbf{v}_1=\left(\begin{array}{l}2 \\t \\3\end{array}\right), \quad \mathbf{v}_2=\left(\begin{array}{l}...
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872
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3 answers
11 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 2
Consider a matrix $A_{4 \times 5}$. Where all the solutions of $A x=0$ has the following form -$$\left[\begin{array}{c}6 c-12 e \\-4 c+10 e \\c \\-5 e \\e\end{array}\righ...
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2 answers
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 3
Let $A$ be a $n \times n$ matrix, and let $u, v, w$ be nonzero vectors in $\mathbf{R}^n$ which are distinct $(\text{so}\; u \neq v, u \neq w$, and $v \neq w).$Suppose $A ...
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 4
Consider a matrix $A_{n \times n}$ having the following characteristic equation -$$\lambda^2(\lambda-3)(\lambda+2)^3(\lambda-4)^3$$What could be $\operatorname{rank}(\mat...
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2 answers
10 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 5
Consider the system below, where $h$ and $k$ are real numbers.$$\begin{array}{r}x+3 y=2 \\3 x-h y=k\end{array}$$Find the values of $h$ and $k$ which give the system infin...
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1 answers
18 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 6
Let $\vec{v}$ be an eigenvector of an invertible matrix $A$. Which of the following are necessarily true?$\vec{v}$ is an eigenvector of $A^{-1}$.$\vec{v}$ is an eigenvect...
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2 answers
32 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 7
Let $\mathrm{A}$ be a $3 \times 3$ matrix. Let $\mathbf{x}, \mathbf{y}, \mathbf{z}$ are linearly independent $3$-dimensional vectors. Suppose that we have$$A \mathbf{x}=\...
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3 answers
14 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 8
Which of the following is/are TRUE?If the echelon form of an $m \times n$ matrix has a pivot in every column then $n \geq m$If the echelon form of an $m \times n$ matrix ...
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4 answers
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 9
$$\left[\begin{array}{ll}3 & 1 \\1 & 0 \\2 & 5\end{array}\right]\left[\begin{array}{lll}a & 1 & 0 \\2 & b & 1\end{array}\right]=A_{3 \times 3}$$At what values of $(a, b),...
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2 answers
10 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 10
The given system has infinitely many solutions for $k=?$ $$\left[\begin{array}{ccc}2 & 2 & -4 \\1 & 3 & -2 \\-4 & k & 8\end{array}\right]\left[\begin{array}{l}x \\y \\z\e...
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2 answers
9 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 11
In the echelon form of the augmented matrix $[A \mid b]:$ A single row of the form $(000 \ldots 0 \mid 0)$ is enough to conclude that the system will have infinitely many...
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4 answers
27 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 12
If $P=\left[\begin{array}{lll}1 & \alpha & 3 \\ 1 & 3 & 3 \\ 2 & 4 & 4\end{array}\right]$ is the adjoint of a $3 \times 3$ matrix $\mathrm{A},$ and $\operatorname{det(A)}...
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 13
Given the following matrix:$$A=\left[\begin{array}{lll}1 & 2 & 2 \\2 & 1 & 2 \\2 & 2 & 1\end{array}\right]$$Consider the following statements:$A^2-4 A-5 I=0$ (where $I$ i...
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3 answers
28 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 14
Mark all the INCORRECT statementsLet $A$ be the matrix of a rotation by angle $30$ degree. That is, for any vector $x,$ the angle between $x$ and $A x$ is always $30$ deg...
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7 answers
19 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 15
Suppose $\mathbf{A}=\mathbf{B C}$, where $\mathbf{B}$ is a $4 \times 2$ matrix and $\mathbf{C}$ is a $2 \times 4$ matrix. Is $\mathbf{A}$ invertible?Yes, $\mathbf{A}$ is ...
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1 answers
8 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 16
Let $A=\left[\begin{array}{cc}1 & 0 \\ -1 & 1 \\ k & 2\end{array}\right]$ and $b=\left[\begin{array}{l}1 \\ 0 \\ 1\end{array}\right]$. For which value of $k$ does the sys...
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967
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1 answers
11 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 17
Let $A$ be an $m \times n$-matrix and let $B$ be an $n \times m$-matrix. Then which of the following statement is not true for all such matrices?$B A$ is definedthe colum...
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686
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1 answers
6 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 18
Consider $A=\left[\begin{array}{ll}1 & 0 \\ 0 & 1 \\ 0 & 2\end{array}\right]$ and $B=\left[\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 2 & 0\end{array}\right]$. Let ...
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1 answers
10 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 19
Let $A$ be a $2 \times 3$-matrix and $b$ a vector in $\mathbb{R}^2$.Consider the following two statements:$(\text{P}1)\; A$ has at most two pivots,$(\text{P2})$ Assuming ...
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3 answers
20 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 20
Suppose $A$ is $3$ by $4,$ and $A x=0$ has all solutions in the following form -$$\mathrm{x}=\mathrm{s}\left[\begin{array}{l}1 \\1 \\1 \\0\end{array}\right]+\mathrm{t}\le...
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2 answers
13 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 21
Consider a matrix $A$ of dimension $m \times n$ such that -$A x=\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$ has no solutions and $A x=\left[\begin{array}{l}0 \\ ...
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3 answers
17 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 22
A three-by-three matrix $B$ is known to have eigenvalues $0,1$ and $2.$ This information is enough to find which one of these (give the answers where possible):The rank o...
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2 answers
28 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 23
Let $A$ be an $m \times n$-matrix. Consider the system of linear equations $A X=b$, which of the following statement is always true:Suppose $m>n$ then rank of augmented m...
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2 answers
29 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 24
Let $A$ be a $3 \times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1,$ and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and ...
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3 answers
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 25
Let $u \in \mathbb{R}^n$ be such that $u^T u=1$ and set $A=u u^T$. What will be the sum of all eigenvalues of $A?$
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GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 26
Given a matrix $A\left(m \times n\right)$ and $A x=b$. Consider below statements :$\text{S}: m<n$$\mathrm{P}:\; A$ has $m \;\mathrm{Linearly ~Independent}$ columns$\text{...
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1 answers
24 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 27
Consider the following set of (column) vectors:$$X=\left\{\left[\begin{array}{l}x_1 \\ x_2 \\ x_3\end{array}\right] \in \mathbb{R}^3 \mid 2 x_1+3 x_2-x_3=0\right\}.$$Whic...
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818
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2 answers
12 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 28
Choose the correct statements.If $b>\frac{2}{3}$, then $A=\left[\begin{array}{ll}2 & b \\ 3 & 1\end{array}\right]$ is invertible.If $P$ is an invertible matrix such that ...
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852
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2 answers
8 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 29
Given matrix $A\left(m \times n\right)$ such that $A x=b.$ Consider below statements :$\text{S1}:$ If $b$ is Linearly Dependent on cols of matrix, then it need not always...
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1.4k
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2 answers
28 votes
GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 30
There exist a $3 \times 3$ real symmetric matrix $\text{S}$ such that -$\text{Statement 1}: \text{S}\left(\begin{array}{l} 1 \\ 2 \\ 3 \end{array}\right)=\left(\begin{arr...
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