Recent questions in Linear Algebra

Consider the matrix $A=\left[\begin{array}{ccc}1 & \sqrt{2} & 0 \\ \sqrt{2} & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$. Then, which of the following statements is/are true?A...

Let $\left[\begin{array}{ccc}l_{11} & 0 & 0 \\ l_{21} & l_{22} & 0 \\ l_{31} & l_{32} & -10\end{array}\right]\left[\begin{array}{ccc}1 & u_{12} & u_{13} \\ 0 & 1 & u_{23}...

​​​​Suppose the polynomial $a+b x+c x^{2}+d x^{3}$ interpolates the data, $(-1,1),(0,3),(1,2)$ and $(2,4)$.Then which one of the following statements is correct?$a=-2 c, ...

Suppose that $2$ is an eigenvalue of the matrix$$\left[\begin{array}{ccc}0 & 3 & -\alpha \\0 & 1 & 0 \\1 & -1 & 3\end{array}\right]$$Then the value of $\alpha$ is equal t...

​For $a, b \in \mathbb{R}$, consider the system of linear equations$$ \begin{array}{r} x+y+a z=2 \\ 2 y+2 z=1 \\ a x+2 z=b \end{array}$$If the system has infinitely many ...

Let $A$ be a $3 \times 3$ matrix whose eigenvalues are $2,3,4$ and let $I$ be the identity matrix of order 3 . If\[A^{-1}=\frac{1}{2 k}\left(A^{2}-9 A\right)+\frac{13}{k}...

Let $P$ and $Q$ be two square matrices of the same order. Then, which of the following matrices is/are necessarily equal to $(P+2 Q)^{2}$ ?$P^{2}+4 P Q+4 Q^{2}$$P(P+2 Q)+...

Let $A$ be a $3 \times 3$ real matrix having eigenvalues $1,2,$ and $3$. If $B=A^{2}+2 A+I$, where $I$ is the $3 \times 3$ identity matrix, then the eigenvalues of $B$ ar...

The second smallest eigenvalue of the eigenvalue problem$\frac{d^{2} y}{d x^{2}}+(\lambda-3) y=0, \quad y(0)=y(\pi)=0$,is$4$$3$$7$$9$

Let $P=\left[\begin{array}{lll}4 & -2 & 2 \\ 6 & -3 & 4 \\ 3 & -2 & 3\end{array}\right]$, and $Q=\left[\begin{array}{lll}3 & -2 & 2 \\ 4 & -4 & 6 \\ 2 & -3 & 5\end{array}...

Let $A=\left[\begin{array}{rrrr}2 & 0 & 1 & 1 \\ 1 & 2 & 5 & -5 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 1 & 3\end{array}\right]$Then the sum of the geometric multiplicities of the di...

Let $A$ and $B$ be $n \times n$ matrices with real entries.Consider the following statements:P: If $A$ is symmetric then $\operatorname{rank}(A)=$ Number of nonzero eigen...