Further Mathematics: Linear Algebra Topical Past Paper Questions

Consider the matrix $$ \boldsymbol{A}=\begin{bmatrix}1&\lambda&-2\\ \lambda&4&1\\ 2&1&\lambda\end{bmatrix}. $$ (a) Show that 3 is the only real value of $\lambda$ for which $A$ is singular. (b) Consider the system of equations $$ \begin{bmatrix}1&3&-2\\ 3&4&1\\ 2&1&3\end{bmatrix}…

[Maximum mark: 17] The matrix M is given by $ M=\begin{bmatrix}1&2&2\\ 3&1&1\\ 2&3&1\end{bmatrix} $. (a) Given that $ M^{3} $ can be written as a quadratic expression in M in the form $ aM^{2} + bM + cI $, determine the values of the constants a, b and c. (b) Show that $ M^{4}=19…

[Maximum mark: 9] Consider the matrix $ M = \begin{bmatrix} 2 & -4 \\ -1 & -1 \end{bmatrix} $. (a) Show that the linear transformation represented by M transforms any point on the line y = x to a point on the same line. [2] (b) Explain what happens to points on the line $ 4y + x…