- \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.
Giving a reason for your answer, explain whether it is possible to evaluate
- \(\mathbf { A B }\)
- \(\mathbf { A } + \mathbf { B }\)
(ii) Given that
$$\left( \begin{array} { r r r }
- 5 & 3 & 1
a & 0 & 0
b & a & b
\end{array} \right) \left( \begin{array} { r r r }
0 & 5 & 0
2 & 12 & - 1
- 1 & - 11 & 3
\end{array} \right) = \lambda \mathbf { I }$$
where \(a , b\) and \(\lambda\) are constants, - determine
- the value of \(\lambda\)
- the value of \(a\)
- the value of \(b\)
- Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1
a & 0 & 0
b & a & b \end{array} \right)\)
(iii) Given that
$$\mathbf { M } = \left( \begin{array} { c c c }
1 & 1 & 1
0 & \sin \theta & \cos \theta
0 & \cos 2 \theta & \sin 2 \theta
\end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$
determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.
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[TURN OVER FOR QUESTION 5]