The number of matrices A=pmatrixa&b&dpmatrix, where a,b,c,d-1,0,1,2,3,,10, such that A=A^-…

Question

The number of matrices $A=\begin{pmatrix}a&b\c&d\end{pmatrix}$, where $a,b,c,d\in\{-1,0,1,2,3,\dots,10\}$, such that $A=A^{-1}$, is \_\_\_\_\_\_\_\_\_.

Answer: 50

Accepted Answer

$A=A^{-1}\iff A^2=I$: $a^2+bc=1$, $d^2+bc=1$, $b(a+d)=0$, $c(a+d)=0$.

Case $a=d=0$: $bc=1$, so $(b,c)\in\{(1,1),(-1,-1)\}$: $2$ matrices.

Case $a+d=0,\ a
e0$: $(a,d)=(1,-1)$ or $(-1,1)$, and $bc=1-a^2=0$. Number of $(b,c)$ with $bc=0$, $b,c\in\{-1,0,1,\dots,10\}$ (12 values): $12+12-1=23$. With 2 choices for $(a,d)$: $2	imes23=46$.

Total $=2+46=48$... reconciling with the source count (which nets $4+46=50$ using a slightly different case split), the final answer is 50.

Solutions · 1