Find non-Zero values of x satisfying the matrix equation, \(x\left[ \begin{matrix} 2x & 2 \\ 3 & x \end{matrix} \right] +2\left[ \begin{matrix} 8 & 5x \\ 4 & 4x \end{matrix} \right] =\left[ \begin{matrix} { x }^{ 2 }+8 & 24 \\ 10 & 6x \end{matrix} \right] \)

If AB = A and BA = B, then show that A² = A and B² = B.

Prove that \(\left| \begin{matrix} 1 & a & { a }^{ 3 } \\ 1 & b & { b }^{ 3 } \\ 1 & c & { c }^{ 3 } \end{matrix} \right| =\left( a-b \right) \left( b-c \right) \left( c-a \right) \left( a+b+c \right) \) 

Prove that \(LHS=\left| \begin{matrix} -{ a }^{ 2 } & ab & ac \\ ab & -{ b }^{ 2 } & bc \\ ac & bc & -{ c }^{ 2 } \end{matrix} \right| ={ 4a }^{ 2 }{ b }^{ 2 }{ c }^{ 2 }\)

Show that \(\left| \begin{matrix} 1 & a & { a } \\ a & 1 & a \\ a & a & 1 \end{matrix} \right| ^{ 2 }=\left| \begin{matrix} 1-2{ a }^{ 2 } & -{ a }^{ 2 } & -{ a }^{ 2 } \\ -{ a }^{ 2 } & -1 & { a }^{ 2 }-2a \\ -{ a }^{ 2 } & { a }^{ 2 }-2a & -1 \end{matrix} \right| \)