Prove that a matrix which is both symmetric as well as skew-symmetric is a null matrix.

If \(A=\left[ \begin{matrix} 3 & -2 \\ 4 & -2 \end{matrix} \right] \), find k so that A² = kA - 2I.

 If \(A=\left[ \begin{matrix} 1 & 2 \\ 2 & 0 \end{matrix} \right] ,B=\left[ \begin{matrix} 3 & -1 \\ 1 & 0 \end{matrix} \right] \) verify the following:

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

Factorise \(\left| \begin{matrix} a & b & c \\ { a }^{ 2 } & { b }^{ 2 } & { c }^{ 2 } \\ bc & ca & ab \end{matrix} \right| \) .