If $A=\left[ {\begin{array}{cc}2&1\\-4&-2\\\end{array}}\right]$ and $I$ is a $2\times 2$identity matrix Find the value of $I+2A+3A^2+....$

Question

If $A=\left[ {\begin{array}{cc}2&1\\-4&-2\\\end{array}}\right]$ and $I=\left[ {\begin{array}{cc}1&0\\0&1\\\end{array}}\right]$ Find the value of $I+2A+3A^2+....$

My answer to this question is

$=\left[ {\begin{array}{cc}5&2\\-8&-3\\\end{array}}\right]$

Am I right?

Answer 1

As $A^2=0$, then $I+2A+3A^2+\cdots=I+2A$ which is what you have as your answer.