Recent questions in Linear Algebra

Consider the matrix $\mathrm{M}=\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 3 & 0 \\ -1 & a & b\end{array}\right]$.Which of the following options is/ are TRUE if $\operatorn...

Consider the matrix $A$ below:$$A=\left[\begin{array}{llll}2 & 3 & 4 & 5 \\0 & 6 & 7 & 8 \\0 & 0 & \alpha & \beta \\0 & 0 & 0 & \gamma\end{array}\right]$$For which of the...

Let $\mathbb{R}$ and $\mathbb{R}^{3}$ denote the set of real numbers and the three dimensional vector space over it, respectively. The value of $\alpha$ for which the set...

​​​​​Consider the matrix $\left[\begin{array}{ll}1 & k \\ 2 & 1\end{array}\right]$, where $k$ is a positive real number. Which of the following vectors is/are eigenvector...

Let $v_1=\left[\begin{array}{l}1 \\ 2 \\ 0\end{array}\right]$ and $v_2=\left[\begin{array}{l}2 \\ 1 \\ 3\end{array}\right]$ be two vectors. The value of the coefficient $...

Let the sets of eigenvalues and eigenvectors of a matrix $B$ be $\left\{\lambda_k \mid 1 \leq k \leq n\right\}$ and $\left\{v_k \mid 1 \leq k \leq n\right\}$, respectivel...

Let $\boldsymbol{x}$ be an $n \times 1$ real column vector with length $l=\sqrt{\boldsymbol{x}^{T} \boldsymbol{x}}$ The trace of the matrix $P=\boldsymbol{x} \boldsymbol{...

$\begin{array}{rlr}a^*=\max_{x, y} & x^2+y^2-8 x+7 \\ \text { s.t. } & \qquad x^2+y^2 \leq 1 \\ & \qquad \qquad y \geq 0\end{array}$Then $a^{\star}$ is$16$$14$$12$$10$Non...