The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are such that \(\mathbf { A } = \left[ \begin{array} { c c } 4 & 2 - 1 & - 3 \end{array} \right]\) and \(\mathbf { B } = \left[ \begin{array} { l l } 4 & 2 2 & 1 \end{array} \right]\).
Explain why \(\mathbf { B }\) has no inverse.
Find the inverse of \(\mathbf { A }\).
Hence, find the matrix \(\mathbf { X }\), where \(\mathbf { A X } = \left[ \begin{array} { c } - 4 1 \end{array} \right]\).
Prove, by mathematical induction, that \(\sum _ { r = 1 } ^ { n } r ( r + 3 ) = \frac { 1 } { 3 } n ( n + 1 ) ( n + 5 )\) for all positive integers \(n\).
A cubic equation has roots \(\alpha , \beta , \gamma\) such that