Recent questions tagged eigen-value

Let $A$ be a 5 x 5 matrix and $\lambda_1$ be one of the eigen value of $A$ such that $\text{rank}(A-\lambda_1I) = 2$, then geometric multiplicity of $\lambda_1$ is ______...

Consider three non-zero matrices $A, ~B$ and $C$ such that $ABB^T = CBB^T$ where $B^T$ is the transpose of $B$. Which of the following statements is necessarily true?$r(A...

Consider non-zero matrices $A\in M_{m\times n}$, $B\in M_{n\times p}$, and $C\in M_{p\times m}$. Then $ABC$, $BCA$, and $CAB$ are all square matrices. Which of the follow...

Suppose that a $4 \times 4$ matrix $A$ has eigenvalue $3$. If$A - 3I = \begin{pmatrix} 1 & -2 & 0 & 4 \\ 0 & 0 & 1 & -3 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix},$t...

The eigenvalues of a $3 \times 3$ matrix $A$ are $-2$, $4$, and $5$. Then the trace of the matrix $A^3-2A^2+3A$ is:$93$ $99$ $112$ $117$

Which one of the following statements is NOT always true for a real square matrix $A$?If $A$ is diagonalizable and all eigenvalues of $A$ are equal to $2$, then $A=2I$. I...

Let $A$ be an $n \times n$ real matrix. Which of the following statement(s) is(are) true?If $A^2=I$ and $A$ has only one eigenvalue, then $A=\pm I$. If $A^3=0$, then $A=0...

If the eigenvalues of a $3 \times 3$ matrix $A$ are $2$, $3$, and $5$, then $A^{-1}$ is$\frac{1}{30}(A^2-10A+31I)$ $\frac{1}{30}(A^2+10A+31I)$ $\frac{1}{30}(A^2-10A-31I)$...

Consider the matrix $A=\begin{bmatrix}4&1&0&0&1\\1&4&1&0&0\\0&1&4&1&0\\0&0&1&4&1\\1&0&0&1&4\end{bmatrix}$.The largest eigenvalue of $A$ is ______.

The eigenvalues of the matrix $A=\begin{pmatrix}2 & 1 & 0\\0 & 2 & 1\\0 & 0 & -3\end{pmatrix}$ are$2,2,-3$ $2,-3,-3$ $1,2,-3$ $2,3,-3$

Two eigenvalues of a $3\times 3$ real matrix $Q$ are $(1+\sqrt{-1})$ and $4$. The determinant of $Q$ is ?

Let $M$ be a real $n\times n$ matrix such that for every non-zero vector $x\in \mathbb{R}^{n},$ we have $x^{T}M x 0.$ ThenSuch an $M$ cannot exist Such $M$s exist and the...

Let $\lambda_1, \lambda_2, \lambda_3$ denote the eigenvalues of the matrix$$A= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos t & \sin t \\ 0 & – \sin t & \cos t \end{bmatrix}.$$...

Let $A$ be a $2 \times 2$ matrix with integer entries. Which of the following could be its eigenvalue?$\sqrt[3]{2}$ $\pi$ $\dfrac{1}{\sqrt{2}}$ $\sqrt{2}$

Let $A$ is a singular matrix of order $5$ and their two eigenvalues are $2+3 \sqrt{-2}, ~2-3 \sqrt{2}$, then which of the following statement(s) is/are true?Trace of $A$ ...

Let $M = \begin{bmatrix} a & b &c \\ b &d & e\\ c & e & f \end{bmatrix}$ be a real matrix with eigenvalues $1, 0$ and $3$. If the eigenvectors corresponding to $1$ and $0...

Let $A$ be the $2 \times 2$ matrix with elements $a_{11} = a_{12} = a_{21} = +1$ and $a_{22} = −1$. Then the eigenvalues of the matrix $A^{19}$ are$1024$ and $−1024 $ $10...

Let $A$ be $3 \times 3$ matrix. Suppose $1$ and $-1$ are two of the three Eigen Values of $A$ and $18$ is one of the Eigen Values of $A^2 + 3A$. ThenBoth $A$ and $A^2 + 3...

Consider the $5\times 5$ matrix below :$\begin{bmatrix} 1&0 &0 &0 &1 \\ 0& 1 & 1 & 1 & 0\\ 0& 1 &1 &1 &0 \\ 0& 1 &1 &1 &0 \\ 1 & 0 & 0 & 0 & 1 \end{bmatrix}$The product ...

Let $A$ be a $3\times 3$ matrix with Eigen values $-1,1,0$. Then $\mid A^{100}+I\mid$ is $\_\_\_\_\_\_\_\_\_\_.$

Consider the following statements :Statement I $:$ Eigenvalues of $M =$ Eigenvalues of $M'$. $(M'$ is the matrix gets after row operation$)$ Stat...

An arbitrary vector $X$ is an eigen vector of the vector of the matrix $A = \begin{pmatrix}1 & 0 & 0 \\0 & a & 0 \\0 & 0 & b\end{pmatrix}$, if $(a, b)=$$(0, 0)$ $(1, 1)$ ...

Consider the following matrix: $$A = \begin{bmatrix} 1.5&0 &1 \\ -0.5 & 0.5& -0.5\\ -0.5& 0 & 0 \end{bmatrix}$$The eigen values of the above matrix are:$-0.5, ~0.5, ~1.5...

A matrix $M$ has Eigen values $1$ & $4$ with Eigen vectors $\begin{pmatrix} 1\\ -1 \end{pmatrix}$ and $\begin{pmatrix} 2\\ 1 \end{pmatrix}$ respectively. The matrix $M$ i...

The Eigen values of a $2 \times 2$ matrix $A$ are $1, -2$, and its Eigen vectors $x_1$ and $x_2$ respectively.The Eigen values and Eigen vectors of the matrix $A^{2} - 3A...

The condition for which the eigenvalues of the matrix $A=\begin{bmatrix} 2 & 1\\ 1 &k \end{bmatrix}$ are positive is$k \frac{1}{2}$ $ k −2$ $ k 0$ $k< \frac{-1}{2}$

Suppose the rank of the matrix$$\begin{pmatrix}1&1&2&2\\1&1&1&3\\a&b&b&1\end{pmatrix}$$is $2$ for some real numbers $a$ and $b$. Then $b$ equals$1$ $3$ $1/2$ $1/3$

If $\mathrm{B}$ has eigenvalues $1, 2, 3, \mathrm{C}$ has eigenvalues $4, 5, 6$, and $\mathrm{D}$ has eigenvalues $7, 8, 9,$ what are the eigenvalues of the $6 \times 6$ ...

Which of the following statements about the eigen values of $I_n$, the $n \times n$ identity matrix (over complex numbers), is true?The eigen values are $1, \omega, \omeg...