matrix

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: $I$ is the $...

Consider the system of linear equations: $Ax = b$, where $A$ is an $n \times n$ matrix, and $x$ and $b$ are $n$-dimensional column vectors. Suppose this system of equations has a u...

Which one of the following matrices has an inverse?

The sum of the eigenvalues of the matrix $A=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}^2$ is ________ (rounded off to the nearest integer).

Let $p$ and $q$ be real numbers such that $p^2+q^2=1$. The eigen values of the matrix $\left[\begin{array}{cc}p & q \\ q & -p\end{array}\right]$ are

M is a 2 x 2 matrix with eigenvalues 4 and 9. The eigenvalues of M² are

The rank of the matrix, M = [0 1 1] [1 0 1], [1 1 0] is _________.

Consider a 2 × 2 matrix M = [v₁ v₂], where, v₁ and v₂ are the column vectors. Suppose M⁻¹ = [u₁ᵀ; u₂ᵀ], where u₁ᵀ and u₂ᵀ are the row vectors. Consider the following statements: St...

The matrix A = $\begin{bmatrix} 3/2 & 0 & 1/2 \ 0 & -1 & 0 \ 1/2 & 0 & 3/2 \end{bmatrix}$ has three distinct eigenvalues and one of its eigenvectors is $\begin{bmatrix} 1 \ 0 \ 1 \...

Consider a non-singular $2 \times 2$ square matrix $\mathbf{A}$. If $trace(\mathbf{A})=4$ and $trace(\mathbf{A}^2)=5$, the determinant of the matrix $\mathbf{A}$ is ________ (up to...

The eigenvalues of the matrix given below are $$\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & -3 & -4 \end{bmatrix}$$

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 $3 \times 3$ identity matrix. The determinant of $B$...

The eigen values of the matrix given below are $$\left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr 0 & { - 3} & { - 4} \cr } } \right]$$

A system matrix is given as follows $$$A = \left[ {\matrix{ 0 & 1 & { - 1} \cr { - 6} & { - 11} & 6 \cr { - 6} & { - 11} & 5 \cr } } \right].$$$ The absolute value of the ratio of...

A matrix has eigen values $$-1$$ and $$-2.$$ The corresponding eigenvectors are $$\left[ {\matrix{ 1 \cr { - 1} \cr } } \right]$$ and $$\left[ {\matrix{ 1 \cr { - 2} \cr } } \right...

An eigen vector of $$p = \left[ {\matrix{ 1 & 1 & 0 \cr 0 & 2 & 2 \cr 0 & 0 & 3 \cr } } \right]$$ is

$${q_1},\,{q_2},{q_3},.......{q_m}$$ are $$n$$-dimensional vectors with $$m < n.$$ This set of vectors is linearly dependent. $$Q$$ is the matrix with $${q_1},\,{q_2},{q_3},..........

For the matrix $$P = \left[ {\matrix{ 3 & { - 2} & 2 \cr 0 & { - 2} & 1 \cr 0 & 0 & 1 \cr } } \right],$$ one of the eigen values is $$-2.$$ Which of the following is an eigen vecto...

Find the eigen values and eigen vectors of the matrix $$\left[ {\matrix{ 3 & { - 1} \cr { - 1} & 3 \cr } } \right]$$

$$A = \left[ {\matrix{ 2 & 0 & 0 & { - 1} \cr 0 & 1 & 0 & 0 \cr 0 & 0 & 3 & 0 \cr { - 1} & 0 & 0 & 4 \cr } } \right].$$ The sum of the eigen values of the matrix $$A$$ is

The rank of the following $$(n+1)$$ $$x$$ $$(n+1)$$ matrix, where $$'a'$$ is a real number is $$$\left[ {\matrix{ 1 & a & {{a^2}} & . & . & . & {{a^n}} \cr 1 & a & {{a^2}} & . & ....

Given the matrix $$A = \left[ {\matrix{ 0 & 1 & 0 \cr 0 & 0 & 1 \cr { - 6} & { - 11} & { - 6} \cr } } \right].\,\,$$ Its eigen values are

$$A$$ $$\,\,5 \times 7$$ matrix has all its entries equal to $$1.$$ Then the rank of a matrix is

The eigen values of the matrix $$\left[ {\matrix{ a & 1 \cr a & 1 \cr } } \right]$$ are