Questions without answers in Engineering Mathematics

Question Description:I am a 3rd-year engineering student preparing for GATE. I have not studied Engineering Mathematics (EM) and Discrete Mathematics (DM) properly yet.I ...

Expected number of tosses to get THT ?

Let $V$ be a subspace of $\mathbb{R}^{10}$. Suppose $A$ is a $10 \times 10$ matrix with real entries. Let $A^k(V) = \{A^k\mathbf{x} : \mathbf{x} \in V\}$ for $k \ge 1$ an...

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Let $A$ be a $2 \times 3$ real matrix whose rows are linearly independent. The dimension of the null space of $A$ is$0$ $1$ $2$ $3$

Let $A=\begin{bmatrix}1 & 0 & 1\\0 & 1 & 1\\0 & 0 & 0\end{bmatrix}$. Consider the system $Ax=y$, where $x,y\in\mathbb{R}^3$.Which of the following vectors $y$ always has ...

Let $$V=\left\{\textbf{x}\in\mathbb{R}^3:x_1+x_2+x_3=0\right\}.$$ Then $V$ is the null space of a matrix of rank$2$ $1$ $3$ $0$

Which of the following subsets of $\mathbb{R}^2$ is a subspace?$\left\{\begin{bmatrix}t\\t^2\end{bmatrix}:t\in\mathbb{R}\right\}$ $\left\{\begin{bmatrix}t\\1\end{bmatrix}...

Let $V=\{\mathbf{v}_1,\mathbf{v}_2\}$ and $W=\{\mathbf{w}_1,\mathbf{w}_2\}$ be two ordered bases of a vector space. Suppose $$\mathbf{w}_1=2\mathbf{v}_1+\mathbf{v}_2,\qqu...

If $A$ is a $2 \times 3$ matrix with rank $2$, then for every $\mathbf{b}\in\mathbb{R}^2$,$A\mathbf{x}=\mathbf{b}$ has no solution. $A\mathbf{x}=\mathbf{b}$ has at least ...

If the system $A\mathbf{x}=\mathbf{b}$ has infinitely many solutions, thenThe nullity of $A$ is $0$. The nullity of $A$ is greater than $0$. $\operatorname{rank}(A)=n$. $...

For which value of $a$ are the vectors$$\mathbf{v}_1=\begin{bmatrix}a\\0\end{bmatrix},\qquad \mathbf{v}_2=\begin{bmatrix}0\\1\end{bmatrix}$$ linearly independent?$a=0$ $a...

Suppose the solution of the homogeneous system $A\mathbf{x}=0$ is$$\mathbf{x}=t\begin{bmatrix}1\\2\\3\end{bmatrix}.$$Then the dimension of the null space of $A$ is$0$ $1$...

Let $$U=\left\{\mathbf{x}\in\mathbb{R}^3:\begin{bmatrix}1 & 1 & 1\end{bmatrix}\mathbf{x}=0\right\}.$$ Which of the following is correct?$U$ is a subspace of $\mathbb{R}^3...

Which of the following is an eigenvector of the matrix$$A=\begin{bmatrix}2 & 0\\0 & 3\end{bmatrix}?$$$\begin{bmatrix}1\\1\end{bmatrix}$ $\begin{bmatrix}0\\1\end{bmatrix}$...

If the characteristic polynomial of a $3 \times 3$ matrix $A$ is $$\lambda^2(\lambda-2),$$ then the rank of $A$ could be$1$ $2$ $3$ $0$

Which of the following is NOT true about an eigenvalue $\lambda$ of a matrix $A$?$A-\lambda I$ is not invertible. $\det(A-\lambda I)=0$. There exists a nonzero vector $\m...

If $V_1$ and $V_2$ are $2$-dimensional subspaces of a $3$-dimensional vector space $V$, then the smallest possible dimension of $V_1\cap V_2$ is$0$ $1$ $2$ $3$

The product of the nonzero eigenvalues of the matrix $$A=\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}$$ is$0$ $1$ $2$ $-1$

If $$A=\begin{bmatrix}1\\2\\3\end{bmatrix}\begin{bmatrix}4 & 5 & 6\end{bmatrix},$$ then $\det(A)$ is$0$ $1$ $24$ $120$

Suppose $A$ and $B$ are diagonalizable matrices. If $A$ and $B$ have the same eigenvalues (counting multiplicities), then$A$ and $B$ are equal. $A$ and $B$ are similar. $...

Suppose $\{\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n\}$ is an orthonormal set in $\mathbb{R}^n$. Which of the following is true?It is linearly dependent. It is a basi...

If $A$ is a symmetric matrix, then the dot product of two eigenvectors corresponding to different eigenvalues is$0$ $1$ $-1$ It depends on the eigenvalues.

Can a matrix with eigenvalues $1, 2, 3$ be similar to the identity matrix $I_3$?Yes No Only if it is invertible Only if it is symmetric

Let $$A=\begin{bmatrix}1 & 0\\0 & -2\end{bmatrix}.$$ The sum of the eigenvalues of $A^6$ is$63$ $64$ $65$ $66$

Consider the matrix $$A=\begin{bmatrix}2 & 3\\x & y\end{bmatrix}.$$ If the eigenvalues of $A$ are $3$ and $4$, then$x=1,\ y=5$ $x=2,\ y=5$ $x=1,\ y=6$ $x=2,\ y=6$

Let $\mathbf{x}$ be a nonzero vector, and define $$A=\mathbf{x}\mathbf{x}^T.$$ When is $A$ an orthogonal matrix?When $\lVert\mathbf{x}\rVert=1$ For every nonzero vector $...

Let $A$ be a real symmetric matrix. Which of the following is always symmetric?$A^2$ $A+I$ $2A$ All of the above

Let $A$ be a real symmetric matrix such that $$\operatorname{trace}(A^T A)=25.$$ If the eigenvalues of $A$ are $3, 4, \lambda$, then $\lambda$ is$0$ $3$ $4$ $5$