Recent questions and answers in Linear Algebra

Two $n \times n$ matrices $A$ and $B$ have a common eigenvalue $2$, and the same corresponding nonzero eigenvector.Which of the following options is/are correct?(Note: $I...

Consider the system of linear equations: $A \boldsymbol{x}=\boldsymbol{b}$, where $A$ is an $n \times n$ matrix, and $\boldsymbol{x}$ and $\boldsymbol{b}$ are $n$-dimensi...

Given a system of equations: $x+2y+2z=b_1$ $5x+y+3z=b_2$ Which of the following is true regarding its ...

Let $v_{1}$ and $v_{2}$ be the two eigenvectors corresponding to distinct eigenvalues of a $3 \times 3$ real symmetric matrix. Which one of the following statements is tr...

Let $\boldsymbol{A}=\left[\begin{array}{ccc}1 & 1 & 1 \\ -1 & -1 & -1 \\ 0 & 1 & -1\end{array}\right]$, and $\boldsymbol{b}=\left[\begin{array}{c}1 / 3 \\ -1 / 3 \\ 0\end...

For a given vector $\mathbf{w}=\left[\begin{array}{lll}1 & 2 & 3\end{array}\right]^{\mathrm{T}}$, the vector normal to the plane defined by $\mathbf{w}^{\mathrm{T}} \math...

The sum of the eigenvalues of the matrix $A=\left[\begin{array}{ll}1 & 2 \\ 3 & 4\end{array}\right]^{2}$ is ____________(rounded off to the nearest integer).

Let $P=\left[\begin{array}{ccc}2 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right]$ and let $I$ be the identity matrix. Then $P^{2}$ is equal to$2 P-I$$P$$I$$P+I$

Consider a matrix $A = \begin{bmatrix} 1 & 0 & 0\\ 0 & 4 & -2\\ 0 & 1 & 1 \end{bmatrix}$.The matrix $A$ satisfies the equation $6A^{-1} = A^{2} + cA + dl$, where $c$ and ...

Let $A= \begin{bmatrix} 1 & 0 & -1 \\ -1 & 2 & 0 \\ 0 & 0 & -2 \end{bmatrix}$ and $B=A^3-A^2-4A+5I$, where $I$ is the $3 \times 3$ identify matrix. The determinant of $B$...

Given that $\textbf{A}= \begin{bmatrix} -5 & -3 \\ 2 & 0 \end{bmatrix}$ and $\textbf{I} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, the value of $A^3$ is$15 \: \text...

​​​​Which one of the following matrices has an inverse?$\left[\begin{array}{ccc}1 & 4 & 8 \\ 0 & 4 & 2 \\ 0.5 & 2 & 4\end{array}\right]$$\left[\begin{array}{lll}1 & 2 & 3...

Let $A$ be a $10\times10$ matrix such that $A^{5}$ is a null matrix, and let $I$ be the $10\times10$ identity matrix. The determinant of $\text{A+I}$ is _________________...

The number of purely real elements in a lower triangular representation of the given $3\times 3$ matrix, obtained through the given decomposition is ______________.$$\beg...

Let $p$ and $q$ be real numbers such that $p^{2}+q^{2}=1$ . The eigenvalues of the matrix $\begin{bmatrix} p & q\\ q& -p \end{bmatrix}$are$1$ and $1$$1$ and $-1$$j$ and $...

The rank of the matrix, $M = \begin{bmatrix} 0 &1 &1 \\ 1& 0 &1 \\ 1& 1 & 0 \end{bmatrix}$, is ______________.

Which one of the following statements is true for all real symmetric matrices?All the eigenvalues are real.All the eigenvalues are positive.All the eigenvalues are distin...