Question

If $I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$ and $E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],$ then $(aI+bE{)}^3=$

• A. $aI+bE$
• B. $a^{3}I+3a^{2}bE$
• C. $a^{3}I+3a{b}^{2}E$
• D. $a^{3}I+b^{3}E$

Answer 1

The correct option is B $a^{3}I+3a^{2}bE$

$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right];E=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$
$aI+bE=\left[\begin{array}{cc}a& 0\\ 0& a\end{array}\right]+\left[\begin{array}{cc}0& b\\ 0& 0\end{array}\right]=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]$
$\Rightarrow (aI+bE{)}^2=\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]=\left[\begin{array}{cc}{a}^2& 2ab\\ 0& a^2\end{array}\right]\Rightarrow (aI+bE{)}^3=\left[\begin{array}{cc}{a}^2& 2ab\\ 0& a^2\end{array}\right]\left[\begin{array}{cc}a& b\\ 0& a\end{array}\right]=\left[\begin{array}{cc}{a}^3& 3a^{2}b\\ 0& a^3\end{array}\right]=\left[\begin{array}{cc}{a}^3& 0\\ 0& a^3\end{array}\right]+\left[\begin{array}{cc}0& 3a^{2}b\\ 0& 0\end{array}\right]=a^3\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]+3a^{2}b\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]=a^{3}I+3a^{2}bE$