Tag: inverse of a matrix and linear equations

If $A$ satisfies the equation $\displaystyle x^{3}-5x^{2}+4x+\lambda =0$, then $\displaystyle A^{-1}$ exists if

If $A$ is an invertiable idempotent matrix and $B=7A^{7}+6A^{6}+5A^{5}+......+A$ then $|B|$ is equal to 

If $\begin{bmatrix} 1 & -1 & x \ 1 & x & 1 \ x & -1 & 1 \end{bmatrix}$ has no inverse, then the real value of $x$ is 

Let p be a nonsingular matrix, and $I + p + p^2 + ..... + p^n = 0$, then find $p^{-1}$.

Matrices A and B satisfy $AB = B^{-1}$, where $ B\quad =\quad \begin{bmatrix} 2 & -1 \ 2 & 0 \end{bmatrix}$, then find without finding $A^{-1}$, the matrix X satisfying $A^{-1}XA = ?$

If $A$ satisfies the equation $x^3-5x^2+4x+kI=0,$ then $A^{-1}$ exists if

If $A^3 = O$, then $I + A + A^2$ equals

If $A$ and $B$ are symmetric matrices and $AB=BA$, then ${ A }^{ -1 }B$ is a

If $A^2 + A - I = 0$, then $A^{-1}$ =

IF $A,B,C$ are non-singular $n\times n$ matrices, then $(ABC)^{-1}$ = ____________.