(i)     Explain why M is a square matrix.

(ii)     Find the set of possible values of det(M).

(i)     Find the value of \(a\).

(ii)     Find the eigenvalues of N.

(iii)     Find corresponding eigenvectors.

(i)     M\(^2 = \) MM only exists if the number of columns of M equals the number of rows of M     R1

hence M is square     AG

(ii)     apply the determinant function to both sides     M1

\(\det (\)M\(^2) = \det (\)M\()\)

use the multiplicative property of the determinant

\(\det (\)M\(^2) = \det (\)M\(){\text{ }}\det (\)M\() = \det (\)M\()\)     (M1)

hence \(\det (\)M\() = 0\) or 1     A1

[4 marks]

(i)     attempt to calculate N\(^2\)        M1

obtain \(\left( {\begin{array}{*{20}{c}} { - {a^2}}&{2{a^2}} \\ { - {a^2}}&{2{a^2}} \end{array}} \right)\)        A1

equating to N        M1

to obtain \(a =  - 1\)        A1

(ii)     N \( = \left( {\begin{array}{*{20}{c}} { - 1}&2 \\ { - 1}&2 \end{array}} \right)\)

N \( - \lambda \)I \( = \left( {\begin{array}{*{20}{c}} { - 1 - \lambda }&2 \\ { - 1}&{2 - \lambda } \end{array}} \right)\)        M1

\(( - 1 - \lambda )(2 - \lambda ) + 2 = 0\)       (A1)

\({\lambda ^2} - \lambda  = 0\)       (A1)

\(\lambda \) is 1 or 0        A1

(iii)     let \(\lambda  = 1\)

to obtain \(\left( {\begin{array}{*{20}{c}} { - 1}&2 \\ { - 1}&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right){\text{ or }}\left( {\begin{array}{*{20}{c}} { - 2}&2 \\ { - 1}&1 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right)\)        M1

hence eigenvector is \(\left( {\begin{array}{*{20}{c}} x \\ x \end{array}} \right)\)        A1

let \(\lambda  = 0\)

to obtain \(\left( {\begin{array}{*{20}{c}} { - 1}&2 \\ { - 1}&2 \end{array}} \right)\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 0 \\ 0 \end{array}} \right)\)        M1

hence eigenvector is \(\left( {\begin{array}{*{20}{c}} {2y} \\ y \end{array}} \right)\)        A1

Note:     Accept specific eigenvectors.

[12 marks]

This was a more successful question for many candidates with a number of fully correct solutions being seen and a significant number of partially correct answers. Most candidates understood what was required from part (a)(i), but part (ii) often resulted in unnecessarily complex algebra which they were unable to manipulate. Part (b) resulted in many wholly successful answers, provided candidates realised the need for care in terms of the manipulation.

This was a more successful question for many candidates with a number of fully correct solutions being seen and a significant number of partially correct answers. Most candidates understood what was required from part (a)(i), but part (ii) often resulted in unnecessarily complex algebra which they were unable to manipulate. Part (b) resulted in many wholly successful answers, provided candidates realised the need for care in terms of the manipulation.