Let M denote the set of all real matrices of order 3 3 and let S=-3,-2,-1,1,2.

Question

Let M denote the set of all real matrices of order $3	imes 3$ and let $S={-3,-2,-1,1,2}$. Let

$\begin{gathered} {S}_{1}={A=[{a}_{ij}]\in M:A={A}^{T}{a}_{ij}\in S,\foralli,j}, \ {S}_{2}={A=[{a}_{ij}]\in M:A=-{A}^{T}{a}_{ij}\in S,\foralli,j}, \ {S}_{3}={A=[{a}_{ij}]\in M:{a}_{11}+{a}_{22}+{a}_{33}=0{a}_{ij}\in S,\foralli,j}. \end{gathered}$

If $n({S}_{1}\cup {S}_{2}\cup {S}_{3})=125\alpha $, then $\alpha $ equls \_\_\_\_\_\_\_\_\_\_.

Answer: 1613

Accepted Answer

$[[|\begin{gathered} {a}_{11} \quad  {a}_{12} \quad  {a}_{13} \ {a}_{21} \quad  {a}_{22} \quad  {a}_{23} \ {a}_{31} \quad  {a}_{32} \quad  {a}_{33} \end{gathered}]]|$

No. of elements in ${S}_{1}:A={A}^{T}\Rightarrow {5}^{3}	imes {5}^{3}$

No. of elements in ${S}_{2}:A=-{A}^{T}\Rightarrow 0$

since no zero in ${S}_{2}$

No. of elements in ${S}_{3}\Rightarrow$

$\begin{gathered} {a}_{11}+{a}_{22}+{a}_{33}=0\Rightarrow (1,2,-3)\Rightarrow 31 \ (1,1,-2)\Rightarrow 3 \ (-1,-1,2)\Rightarrow 3 \end{gathered}\Rightarrow 12	imes {5}^{6}$

$\begin{gathered} n({S}_{1}\cap {S}_{3})=12	imes {5}^{3} \ n({S}_{1}\cup {S}_{2}\cup {S}_{3})={5}^{6}(1+12)-12	imes {5}^{3} \ \, \Rightarrow {5}^{3}	imes [13	imes {5}^{3}-12]=125\alpha \ \, \alpha =1613 \end{gathered}$

Solutions · 1