Recent questions tagged matrices

A sequence $x[n]$ is specified as $\begin{bmatrix}x[n] \\x[n – 1]\end{bmatrix}=\begin{bmatrix}1&1\\1&0\end{bmatrix}^n\begin{bmatrix}1\\0\end{bmatrix}$,for $n \geq 2$.The ...

For $A = \begin{bmatrix} 1 &\tan x \\ -\tan x &1 \end{bmatrix},$ the determinant of $A^{T}A^{-1}$ is$\sec^{2}x$$\cos 4x$$1$$0$

The value of $x$ for which all the eigen-values of the matrix given below are real is $$\begin{bmatrix} 10&5+j &4 \\ x&20 &2 \\4 &2 &-10 \end{bmatrix}$$$5+j$$5-j$$1-5j$$1...

The state variable representation of a system is given as$\dot{x} = \begin{bmatrix} 0 &1 \\ 0 &-1 \end{bmatrix}\: ; x(0)=\begin{bmatrix} 1\\0 \end{bmatrix}$$y=\begin{bm...

Two sequences $\begin{bmatrix}a, & b, & c \end{bmatrix}$ and $\begin{bmatrix}A, & B, & C \end{bmatrix}$ are related as,$$\begin{bmatrix}A \\ B \\ C \end{bmatrix} = \be...

The state transition matrix $\phi(t)$ of a system $\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \...

Which one of the following statements is NOT true for a square matrix $A$?If $A$ is upper triangular, the eigenvalues of $A$ are the diagonal elements of itIf $A$ is real...

The state equation of a second-order linear system is given by$$\dot{x}(t)=Ax(t), \:\:\:\:\:\:\:\:x(0)=x_{0}$$For $x_{0}= \begin{bmatrix} 1\\ -1 \end{bmatrix},$ $x(t)...

The determinant of matrix $A$ is $5$ and the determinant of matrix B is $40$. The determinant of matrix $AB$ is ________

The system of linear equations $\begin{pmatrix} 2 & 1 & 3\\ 3&0 &1 \\ 1& 2 &5 \end{pmatrix} \begin{pmatrix} a\\ b\\ c \end{pmatrix} = \begin{pmatrix} 5\\ -4\\ 14 \end{pma...

The maximum value of the determinant among all $2 \times 2$ real symmetric matrices with trace $14$ is __________.

A real $(4 \times 4)$ matrix $A$ satisfies the equation $A^{2} = I$, where $I$ is the $(4 \times 4)$ identity matrix. The positive eigen value of $A$ is ______.

Consider the matrix $$J_{6} = \begin{bmatrix} 0&0 &0 &0 &0 &1 \\ 0& 0& 0& 0& 1&0 \\ 0& 0& 0& 1& 0&0 \\ 0&0 & 1& 0&0 &0 \\0 &1 &0 &0 &0 &0 \\1 &0 &0 &0 & 0& 0\end{bmatrix}...

Let $A$ be an $m \times n$ matrix and $B$ an $n \times m$ matrix. It is given that determinant $(I_{m} + AB) =$ determinant $(I_{n} + BA),$ where $I_{k}$ is the $k \time...

Let $\text{M}$ be a real $4\times 4$ matrix. Consider the following statements:$S1: M $ has $4$ linearly independent eigenvectors.$S2: M$ has $4$ distinct eigenvalues. $S...

The rank of the matrix $\begin{bmatrix} 1 & -1& 0& 0& 0& \\ 0& 0& 1& -1& 0& \\ 0& 1& -1& 0& 0& \\ -1& 0& 0& 0& 1& \\ 0& 0&...

The rank of the matrix $$\textbf{M} = \begin{bmatrix} 5&10&10 \\ 1 &0 &2 \\ 3&6&6 \end{bmatrix}$$ is$0$$1$$2$$3$

Consider the 5 $\times$ 5 matrix$$\begin{bmatrix} 1&2&3&4&5\\ 5&1 &2& 3 &4\\ 4&5&1&2&3\\ 3&4&5&1&2\\ 2&3&4&5&1 \end{bmatrix}$$It is given that A has only one real eigenv...