Eigenvalues and Characteristic Equation

If  $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$  then what would be the value of  $A^{-1}$

Let $A$ be a $2\times 2$ matrix with real entries such that zero is the only solution of the equation $det\left(A-xI\right)=0$. Then

Find the eigenvalues and eigenvectors for the matrix $\left(\begin{array}{ll}3& 8\\ 0& 2\end{array}\right)$.

Show that the characteristic equation for the matrix $\left(\begin{array}{rrr}5& 4& 1\\ -6& -2& 3\\ 8& 8& 3\end{array}\right)$ is ${\lambda }^3-6{\lambda }^2-9\lambda +14=0$.

Find the eigenvalues and corresponding eigenvectors for the matrix $\left(\begin{array}{rr}1& 3\\ 5& -1\end{array}\right)$.

Find the eigenvalues and corresponding eigenvectors for the matrix $\left(\begin{array}{ll}3& 4\\ 1& 0\end{array}\right)$.

Show that if ${\alpha }_1, {\alpha }_2,\dots \dots \dots ,{\alpha }_n$ are $n$ characteristic roots of a square matrix $A$ of order $n,$ then the roots of the matrix $A^2$ be ${\alpha }_1^2, {\alpha }_2^2,\dots \dots \dots {\alpha }_n^2$

Prove that the two matrices $A$ and $P^{-1}AP$ have the same characteristic roots and hence show that square matrices $AB \& BA$ have same characteristic roots if at least one of them is invertible.

The smallest and largest eigenvalues of the following matrix are:

$\left[\begin{array}{ccc}3& -2& 2\\ 4& -4 & 6\\ 2& -3& 5\end{array}\right]$

For the matrix $A=\left[\begin{array}{ll}1& 3\\ 2& 4\end{array}\right]$, find $a+b$ such that $A^2+aA+bI=0$.

Cotyledons are also called-

If $A=\left[\begin{array}{ccc}1& 0& 1\\ 1& 1& 0\\ 0& 1& 0\end{array}\right] \text{and} I=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],$ then $A^{-1}$ equals

If $A^2-A+I=O,$ then the inverse of A is

If $A^2-A+I=0$, then the inverse of $A$ is

If ${\mathrm{A}}^3=\mathrm{ }\mathrm{O}$ , then $\mathrm{I}+\mathrm{A}+{\mathrm{A}}^2$ equals

If $A^2-A+I=0,$then the inverse of $A$ is

The characteristic roots of the matrix$\left[\begin{array}{c}1 0 0\\ 2 3 0\\ 4 5 6\end{array}\right]$are

If $A=\left[\begin{array}{cc}3& 1\\ -1& 2\end{array}\right]$ and$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, then the correct option is