An algebra problem by Anik Saha

The given matrix is a rank 1 matrix as the 1st row is twice the second row. The characteristic equation of this matrix is:

$\lambda^2 - 7 \lambda = 0$

The eigenvalues are therefore $7$ and $0$ . Using this information, the matrix A can be factorised as such:

$A = V \left[\begin{matrix} 7&0\\0&0\end{matrix}\right] V^{-1}$ $A = VDV^{-1}$ $A^{100} = ( VDV^{-1})( VDV^{-1}) \dots (VDV^{-1})$ $A^{100 }= VD^{100}V^{-1}$ $A^{100} = V \left[\begin{matrix} 7^{100}&0\\0&0\end{matrix}\right] V^{-1}$ $A^{100 }= 7^{99}V \left[\begin{matrix} 7&0\\0&0\end{matrix}\right] V^{-1}$ $A^{100} = 7^{99}A$