Question

If $A=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ and $B=\left[\begin{array}{ccc}-4& -3& -3\\ 1& 0& 1\\ 4& 4& 3\end{array}\right]$ are two matrices, then the value of the determinant $\left(A+A^2{B}^2+A^3+A^4{B}^4+.........20terms\right)$ is

• A. ${\left(20\right)}^3$
• B. $2{\left(20\right)}^3$
• C. $-{\left(20\right)}^3$
• D. $0$

Answer 1

Answer: (D)

$A^2=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]$ $=\left[\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right]=A$
$B^2=\left[\begin{array}{ccc}-4& -3& -3\\ 1& 0& 1\\ 4& 4& 3\end{array}\right]\left[\begin{array}{ccc}-4& -3& -3\\ 1& 0& 1\\ 4& 4& 3\end{array}\right]$ $=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]=I$
$B^2=B^4=B^6=B^8=.......=B^{20}=I$
$A^2=A^3=A^4=.......A^{20}=A$
So, $\left|A+A^2{B}^2+A^3.......+A^{20}{B}^{20}\right|$ $=\left|A+A+A.......+A\right|=\left|20A\right|$ $={\left(20\right)}^3\left|A\right|$ ....(1)
$\left|A\right|=\left|\begin{array}{ccc}2& -2& -4\\ -1& 3& 4\\ 1& -2& -3\end{array}\right|=2\left(-9+8\right)+2\left(3-4\right)-4\left(2-3\right)$
$=-2-2+4=0$
Hence, from equation (1),
$\left|A+A^2{B}^2+A^3.......+A^{20}{B}^{20}\right|=0$