Recent questions in Linear Algebra

Let $\mathbf{A}$ and $\mathbf{B}$ be real symmetric matrices of same size. Which one of the following options is correct?$\mathbf{A}^{\mathrm{T}}=\mathrm{A}^{-1}$$\mathbf...

​​​​​Consider the system of linear equations\[\begin{array}{c}x+2 y+z=5 \\2 x+a y+4 z=12 \\2 x+4 y+6 z=b\end{array}\]The values of $a$ and $b$ such that there exists a no...

​​​​The matrix $\left[\begin{array}{ll}1 & a \\ 8 & 3\end{array}\right]$ (where $a>0$ ) has a negative eigenvalue if $a$ is greater than$\frac{3}{8}$$\frac{1}{8}$$\frac{1...

If the sum and product of eigenvalues of a $2 \times 2$ real matrix $\begin{bmatrix} 3 & p\\ p & q \end{bmatrix}$ are $4$ and $-1$ respectively, then $\left | p \right |$...

Consider an $n \times n$ matrix $\text{A}$ and a non-zero $n \times 1$ vector $p.$ Their product $Ap=\alpha ^{2}p$, where $\alpha \in \Re$ and $\alpha \notin \left \{ -1...

A matrix $P$ is decomposed into its symmetric part $S$ and skew symmetric part $V$. If $$S= \begin{pmatrix} -4 & 4 & 2 \\ 4 & 3 & 7/2 \\ 2 & 7/2 & 2 \end{pmatrix}, \: \: ...

In matrix equation $[A] \{X\}=\{R\}$,$[A] = \begin{bmatrix} 4 & 8 & 4 \\ 8 & 16 & -4 \\ 4 & -4 & 15 \end{bmatrix} \{X\} = \begin{Bmatrix} 2 \\ 1 \\ 4 \end{Bmatrix} \text{...

Consider the matrix$$P=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}$$The number of distinct eigenvalues$0$$1$$2$$3$