9 It is given that \(\mathbf{e}\) is an eigenvector of the matrix \(\mathbf{A}\), with corresponding eigenvalue \(\lambda\).
(i) Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{2}\), with corresponding eigenvalue \(\lambda^{2}\).

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are given by
\(\mathbf{A}=\left(\begin{array}{ccc}
n & 1 & 3 \\
0 & 2 n & 0 \\
0 & 0 & 3 n
\end{array}\right) \quad \text { and } \quad \mathbf{B}=(\mathbf{A}+n \mathbf{I})^{2}\)
where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix and \(n\) is a non-zero integer.
(ii) Find, in terms of \(n\), a non-singular matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{B}=\mathbf{P D P}^{-1}\).

11 Answer only one of the following two alternatives.

EITHER

A \(3\times3\) matrix \(A\) has distinct eigenvalues \(2\), \(1\), \(3\), with corresponding eigenvectors \(\begin{pmatrix}1\\1\\0\end{pmatrix}\), \(\begin{pmatrix}-1\\0\\b\end{pmatrix}\), \(\begin{pmatrix}0\\1\\-1\end{pmatrix}\), respectively, where \(b\) is a positive constant.

(i) Find \(A\) in terms of \(b\).

(ii) Find \(A^{-1}\begin{pmatrix}0\\2\\-2\end{pmatrix}\).

(iii) It is given that \(A^n\begin{pmatrix}1\\1\\0\end{pmatrix}=\begin{pmatrix}4\\4\\0\end{pmatrix}\) and \(A^n\begin{pmatrix}-1\\0\\b\end{pmatrix}=\begin{pmatrix}-1\\0\\b^{-1}\end{pmatrix}\). Find the values of \(n\) and \(b\).

It is given that \(\mathbf{e}\) is an eigenvector of the matrix \(\mathbf{A}\) with corresponding eigenvalue \(\lambda\).
(i) Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{3}\) and state the corresponding eigenvalue.

It is given that
\(\mathbf{A}=\left(\begin{array}{rr}
2 & 0 \\
-1 & 3
\end{array}\right) .\)
(ii) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that
\(\mathbf{A}^{3}+\mathbf{I}=\mathbf{P D P} \mathbf{P}^{-1}\)
where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix.

It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf{A}\) with \(\mathbf{e}\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf{B}\) for which \(\mathbf{e}\) is also a corresponding eigenvector.
(i) Show that \(\lambda+\mu\) is an eigenvalue of the matrix \(\mathbf{A}+\mathbf{B}\) with \(\mathbf{e}\) as a corresponding eigenvector.

The matrix \(\mathbf{A}\), given by
\(\mathbf{A}=\left(\begin{array}{rrr}
2 & 0 & 1 \\
-1 & 2 & 3 \\
1 & 0 & 2
\end{array}\right)\)
has \(\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ 4 \\ -1\end{array}\right)\) and \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)\) as eigenvectors.
(ii) Find the corresponding eigenvalues.

The matrix \(\mathbf{B}\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ 4 \\ -1\end{array}\right)\) and \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)\) respectively.
(iii) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A}+\mathbf{B})^{3}=\mathbf{P D P}^{-1}\).

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -11 & 1 & 8 \\ 0 & -2 & 0 \\ -16 & 1 & 13 \end{array}\right)\)
(a) Show that \(\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)\) is an eigenvector of \(\mathbf{A}\) and state the corresponding eigenvalue.
(b) Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^{3}-19 \lambda-30=0\) and hence find the other eigenvalues of \(\mathbf{A}\).
(c) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).

The matrix \(\mathbf{P}\) is given by
\(\mathbf{P}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ 0 & 2 & 1 \\ 0 & 0 & -1 \end{array}\right) .\)
(a) State the eigenvalues of \(\mathbf{P}\).

(b) Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\).

The \(3 \times 3\) matrix \(\mathbf{A}\) has distinct non-zero eigenvalues \(a, \frac{1}{2}, 2\) with corresponding eigenvectors
\(\left(\begin{array}{l} 1 \\ 0 \\ \end{array}\right), \quad\left(\begin{array}{r} -1 \\ 2 \\ \end{array}\right), \quad\left(\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right),\)
respectively.
(c) Find \(\mathbf{A}^{-1}\) in terms of \(a\).

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{lll} 6 & -9 & 5 \\ 5 & -8 & 5 \\ 1 & -1 & 2 \end{array}\right) .\)
(a) Find the eigenvalues of \(\mathbf{A}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to show that \(\mathbf{A}^{-1}=p \mathbf{A}^{2}+q \mathbf{I}\), where \(p\) and \(q\) are constants to be determined.

It is given that
\(\mathbf{A}=\left(\begin{array}{rrr} 2 & 3 & 1 \\ 0 & -2 & 1 \\ 0 & 0 & 1 \end{array}\right) .\)
(i) Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\left(\begin{array}{l}1 \\ 0 \\ 0\end{array}\right)\).

(ii) Write down the negative eigenvalue of \(\mathbf{A}\) and find a corresponding eigenvector.

(iii) Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A}+\mathbf{A}^{6}\).

The matrix \(\mathbf{A}\), given by
\(\mathbf{A}=\left(\begin{array}{lll} 1 & 2 & -2 \\ 6 & 4 & -6 \\ 6 & 5 & -7 \end{array}\right),\)
has eigenvalues \(1,-1\) and -2 .
(i) Find a set of corresponding eigenvectors.

(ii) The matrix \(\mathbf{B}\) is given by \(\mathbf{B}=\mathbf{A}-2 \mathbf{I}\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. Write down the eigenvalues of \(\mathbf{B}\), and state a set of corresponding eigenvectors.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{lll} 6 & -8 & 7 \\ 7 & -9 & 7 \\ 6 & -6 & 5 \end{array}\right)\)
(i) Given that \(\left(\begin{array}{l}1 \\ 1 \\ 0\end{array}\right)\) is an eigenvector of \(\mathbf{A}\), find the corresponding eigenvalue.

(ii) Given also that -1 is an eigenvalue of \(\mathbf{A}\), find a corresponding eigenvector.

(iii) It is given that the determinant of \(\mathbf{A}\) is equal to the product of the eigenvalues of \(\mathbf{A}\). Use this result to find the third eigenvalue of \(\mathbf{A}\), and find also a corresponding eigenvector.
(iv) Write down matrices \(\mathbf{P}\) and \(\mathbf{D}\) such that \(\mathbf{P}^{-1} \mathbf{A P}=\mathbf{D}\), where \(\mathbf{D}\) is a diagonal matrix, and hence find the matrix \(\mathbf{A}^{n}\) in terms of \(n\), where \(n\) is a positive integer.

It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf{A}\), with corresponding eigenvector \(\mathbf{e}\). Show that \(\lambda^{-1}\) is an eigenvalue of \(\mathbf{A}^{-1}\) for which \(\mathbf{e}\) is a corresponding eigenvector.

Deduce that \(\lambda+\lambda^{-1}\) is an eigenvalue of \(\mathbf{A}+\mathbf{A}^{-1}\).

It is given that \(1\) is an eigenvalue of the matrix \(\mathbf{A}\), where

\(\mathbf{A}=\begin{pmatrix}2&0&1\\-1&2&3\\1&0&2\end{pmatrix}.\)

Find a corresponding eigenvector.

It is also given that \(\begin{pmatrix}0\\1\\0\end{pmatrix}\) and \(\begin{pmatrix}1\\2\\1\end{pmatrix}\) are eigenvectors of \(\mathbf{A}\). Find the corresponding eigenvalues.

Hence find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\left(\mathbf{A}+\mathbf{A}^{-1}\right)^3=\mathbf{PDP}^{-1}\).

The matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrr} -2 & 2 & 2 \\ 2 & 1 & 2 \\ -3 & -6 & -7 \end{array}\right),\)
has an eigenvector \(\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right)\). Find the corresponding eigenvalue.

It is given that if the eigenvalues of a general \(3 \times 3\) matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right),\)
are \(\lambda_{1}, \lambda_{2}\) and \(\lambda_{3}\) then
\(\lambda_{1}+\lambda_{2}+\lambda_{3}=a+e+i\)
and
the determinant of \(\mathbf{A}\) has the value \(\lambda_{1} \lambda_{2} \lambda_{3}\).
Use these results to find the other two eigenvalues of the matrix \(\mathbf{M}\), and find corresponding eigenvectors.

The matrix A, where
\(\mathbf{A}=\left(\begin{array}{rrr} 1 & 0 & 0 \\ 10 & -7 & 10 \\ 7 & -5 & 8 \end{array}\right),\)
has eigenvalues 1 and 3 . Find corresponding eigenvectors.

It is given that \(\left(\begin{array}{l}0 \\ 2 \\ 1\end{array}\right)\) is an eigenvector of \(\mathbf{A}\). Find the corresponding eigenvalue.

Find a diagonal matrix \(\mathbf{D}\) and matrices \(\mathbf{P}\) and \(\mathbf{P}^{-1}\) such that \(\mathbf{P}^{-1} \mathbf{A P}=\mathbf{D}\).

OR

One of the eigenvalues of the matrix \(\mathbf M\), where

\(\mathbf M=\begin{pmatrix}3&-4&2\\-4&\alpha&6\\2&6&-2\end{pmatrix}\),

is \(-9\). Find the value of \(\alpha\).

Find

(i) the other two eigenvalues, \(\lambda_1\) and \(\lambda_2\), of \(\mathbf M\), where \(\lambda_1\gt\lambda_2\),

(ii) corresponding eigenvectors for all three eigenvalues of \(\mathbf M\).

It is given that \(\mathbf x=a\mathbf e_1+b\mathbf e_2\), where \(\mathbf e_1\) and \(\mathbf e_2\) are eigenvectors of \(\mathbf M\) corresponding to \(\lambda_1\) and \(\lambda_2\), respectively. Show that \(\mathbf M\mathbf x=p\mathbf e_1+q\mathbf e_2\), expressing \(p\) and \(q\) in terms of \(a\) and \(b\).

Find a matrix \(\mathbf{A}\) whose eigenvalues are \(-1,1,2\) and for which corresponding eigenvectors are
\(\left(\begin{array}{l} 1 \\ 0 \\ \end{array}\right), \quad\left(\begin{array}{l} 1 \\ 1 \\ \end{array}\right), \quad\left(\begin{array}{l} 0 \\ 1 \\ \end{array}\right),\)
respectively.

EITHER

The vector \(\mathbf e\) is an eigenvector of the matrix \(\mathbf A\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf B\), with corresponding eigenvalue \(\mu\).

(i) Show that \(\mathbf e\) is an eigenvector of the matrix \(\mathbf{AB}\) with corresponding eigenvalue \(\lambda\mu\).

(ii) Find the eigenvalues and corresponding eigenvectors of

\(\mathbf A=\begin{pmatrix}0&1&-3\\4&-3&-2\\1&1&2\end{pmatrix}.\)

(iii) The matrix

\(\mathbf B=\begin{pmatrix}3&6&1\\1&-2&-1\\6&6&-2\end{pmatrix}\)

has eigenvectors \(\begin{pmatrix}1\\-1\\0\end{pmatrix}\), \(\begin{pmatrix}1\\-1\\1\end{pmatrix}\) and \(\begin{pmatrix}1\\0\\1\end{pmatrix}\). Find the eigenvalues of \(\mathbf{AB}\), and state the corresponding eigenvectors.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{lll} 4 & -5 & 3 \\ 3 & -4 & 3 \\ 1 & -1 & 2 \end{array}\right)\)

Show that \(\mathbf{e}=\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right)\) is an eigenvector of \(\mathbf{A}\) and state the corresponding eigenvalue.

Find the other two eigenvalues of \(\mathbf{A}\).

The matrix \(\mathbf{B}\) is given by
\(\mathbf{B}=\left(\begin{array}{rrr} -1 & 4 & 0 \\ -1 & 3 & 1 \\ 1 & -1 & 3 \end{array}\right)\)

Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{B}\) and deduce an eigenvector of the matrix \(\mathbf{A B}\), stating the corresponding eigenvalue.

The square matrix \(\mathbf{A}\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf{e}\). The non-singular matrix \(\mathbf{M}\) is of the same order as \(\mathbf{A}\). Show that \(\mathbf{M e}\) is an eigenvector of the matrix \(\mathbf{B}\), where \(\mathbf{B}=\mathbf{M} \mathbf{A} \mathbf{M}^{-1}\), and that \(\lambda\) is the corresponding eigenvalue.

Let
\(\mathbf{A}=\left(\begin{array}{rrr} -1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array}\right)\)

Write down the eigenvalues of \(\mathbf{A}\) and obtain corresponding eigenvectors.

Given that
\(\mathbf{M}=\left(\begin{array}{lll} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right)\)
find the eigenvalues and corresponding eigenvectors of \(\mathbf{B}\).

The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{2}\) and state the corresponding eigenvalue.

Find the eigenvalues of the matrix \(\mathbf{B}\), where
\(\mathbf{B}=\left(\begin{array}{lll} 1 & 3 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 2 \end{array}\right) .\)

Find the eigenvalues of \(\mathbf{B}^{4}+2 \mathbf{B}^{2}+3 \mathbf{I}\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix.

The square matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{2}\) and state the corresponding eigenvalue.

Find the eigenvalues of the matrix \(\mathbf{B}\), where
\(\mathbf{B}=\left(\begin{array}{lll} 1 & 3 & 0 \\ 2 & 0 & 2 \\ 1 & 1 & 2 \end{array}\right) .\)

Find the eigenvalues of \(\mathbf{B}^{4}+2 \mathbf{B}^{2}+3 \mathbf{I}\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix.

Answer only one of the following two alternatives.

EITHER
Let \(I_{n}=\int_{0}^{1}\left(1+x^{2}\right)^{n} \mathrm{~d} x\). Show that, for all integers \(n\),
\((2 n+1) I_{n}=2 n I_{n-1}+2^{n} .\)

Evaluate \(I_{0}\) and hence find \(I_{3}\).

Given that \(I_{-1}=\frac{1}{4} \pi\), find \(I_{-3}\).

OR

The vector \(\mathbf{e}\) is an eigenvector of each of the \(3 \times 3\) matrices \(\mathbf{A}\) and \(\mathbf{B}\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Justifying your answer, state an eigenvalue of \(\mathbf{A}+\mathbf{B}\).

The matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrr} 6 & -1 & -6 \\ 1 & 0 & -2 \\ 3 & -1 & -3 \end{array}\right),\)
has eigenvectors \(\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)\). Find the corresponding eigenvalues.

The matrix \(\mathbf{B}\), where
\(\mathbf{B}=\left(\begin{array}{rrr} 8 & -2 & -8 \\ 2 & 0 & -4 \\ 4 & -2 & -4 \end{array}\right),\)
also has eigenvectors \(\left(\begin{array}{l}1 \\ 1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)\), for which \(-2,2,4\), respectively, are corresponding eigenvalues. The matrix \(\mathbf{M}\) is given by \(\mathbf{M}=\mathbf{A}+\mathbf{B}-5 \mathbf{I}\), where \(\mathbf{I}\) is the \(3 \times 3\) identity matrix. State the eigenvalues of \(\mathbf{M}\).

Find matrices \(\mathbf{R}\) and \(\mathbf{S}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{M}^{5}=\mathbf{R D S}\).
[You should show clearly all the elements of the matrices \(\mathbf{R}, \mathbf{S}\) and \(\mathbf{D}\).]

Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf{A}\) with \(\mathbf{e}\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf{B}\) for which \(\mathbf{e}\) is also a corresponding eigenvector, then \(\lambda+\mu\) is an eigenvalue of the matrix \(\mathbf{A}+\mathbf{B}\) with \(\mathbf{e}\) as a corresponding eigenvector.

The matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & -1 & 0 \\ -4 & -6 & -6 \\ 5 & 11 & 10 \end{array}\right)\)
has \(\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right)\) as an eigenvector. Find the corresponding eigenvalue.

The other two eigenvalues of \(\mathbf{A}\) are 1 and 2, with corresponding eigenvectors \(\left(\begin{array}{r}1 \\ 2 \\ -3\end{array}\right)\) and \(\left(\begin{array}{r}1 \\ 1 \\ -2\end{array}\right)\) respectively. The matrix \(\mathbf{B}\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left(\begin{array}{r}1 \\ -1 \\ 1\end{array}\right),\left(\begin{array}{r}1 \\ 2 \\ -3\end{array}\right)\), \(\left(\begin{array}{r}1 \\ 1 \\ -2\end{array}\right)\) respectively. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A}+\mathbf{B})^{4}=\mathbf{P D P} \mathbf{P}^{-1}\).
[You are not required to evaluate \(\mathbf{P}^{-1}\).]

The matrix \(\mathbf{A}\) has \(\lambda\) as an eigenvalue with \(\mathbf{e}\) as a corresponding eigenvector. Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{2}\) and state the corresponding eigenvalue.

Given that one eigenvalue of \(\mathbf{A}\) is 3 , find an eigenvalue of the matrix \(\mathbf{A}^{4}+3 \mathbf{A}^{2}+2 \mathbf{I}\), justifying your answer.

Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}=\left(\begin{array}{rrr}4 & -1 & 1 \\ -1 & 0 & -3 \\ 1 & -3 & 0\end{array}\right)\).

Find a non-singular matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).

The matrix \(\mathbf{A}\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf{e}\). Prove that the matrix \((\mathbf{A}+k \mathbf{I})\), where \(k\) is a real constant and \(\mathbf{I}\) is the identity matrix, has an eigenvalue ( \(\lambda+k\) ) with corresponding eigenvector \(\mathbf{e}\).

The matrix \(\mathbf{B}\) is given by
\(\mathbf{B}=\left(\begin{array}{rrr} 2 & 2 & -3 \\ 2 & 2 & 3 \\ -3 & 3 & 3 \end{array}\right) .\)

Two of the eigenvalues of \(\mathbf{B}\) are -3 and 4 . Find corresponding eigenvectors.

Given that \(\left(\begin{array}{r}1 \\ -1 \\ -2\end{array}\right)\) is an eigenvector of \(\mathbf{B}\), find the corresponding eigenvalue.

Hence find the eigenvalues of \(\mathbf{C}\), where
\(\mathbf{C}=\left(\begin{array}{rrr} -1 & 2 & -3 \\ 2 & -1 & 3 \\ -3 & 3 & 0 \end{array}\right),\)
and state corresponding eigenvectors.

The matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrr} -2 & 2 & 2 \\ 2 & 1 & 2 \\ -3 & -6 & -7 \end{array}\right),\)
has an eigenvector \(\left(\begin{array}{r}0 \\ 1 \\ -1\end{array}\right)\). Find the corresponding eigenvalue.

It is given that if the eigenvalues of a general \(3 \times 3\) matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right),\)
are \(\lambda_{1}, \lambda_{2}\) and \(\lambda_{3}\) then
\(\lambda_{1}+\lambda_{2}+\lambda_{3}=a+e+i\)
and
the determinant of \(\mathbf{A}\) has the value \(\lambda_{1} \lambda_{2} \lambda_{3}\).
Use these results to find the other two eigenvalues of the matrix \(\mathbf{M}\), and find corresponding eigenvectors.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{array}\right) .\)

Given that one eigenvector of \(\mathbf{A}\) is \(\left(\begin{array}{r}1 \\ -2 \\ -1\end{array}\right)\), find the corresponding eigenvalue.

Given also that another eigenvalue of \(\mathbf{A}\) is 4, find a corresponding eigenvector.

Given further that \(\left(\begin{array}{r}1 \\ -4 \\ -1\end{array}\right)\) is an eigenvector of \(\mathbf{A}\), with corresponding eigenvalue 1 , find matrices \(\mathbf{P}\) and \(\mathbf{Q}\), together with a diagonal matrix \(\mathbf{D}\), such that \(\mathbf{A}^{5}=\mathbf{P D Q}\).

Answer only one of the following two alternatives.
EITHER

A \(3 \times 3\) matrix \(\mathbf{A}\) has eigenvalues \(-1,1,2\), with corresponding eigenvectors
\(\left(\begin{array}{r} 0 \\ 1 \\ -1 \end{array}\right), \quad\left(\begin{array}{r} -1 \\ 0 \\ \end{array}\right), \quad\left(\begin{array}{l} 1 \\ 1 \\ \end{array}\right),\)
respectively. Find
(i) the matrix \(\mathbf{A}\),
(ii) \(\mathbf{A}^{2 n}\), where \(n\) is a positive integer.

OR

Determine the rank of the matrix
\(\mathbf{A}=\left(\begin{array}{llll} 1 & -1 & -1 & 1 \\ 2 & -1 & -4 & 3 \\ 3 & -3 & -2 & 2 \\ 5 & -4 & -6 & 5 \end{array}\right)\)

Show that if
\(\mathbf{A x}=p\left(\begin{array}{l} 1 \\ 2 \\ 3 \\ \end{array}\right)+q\left(\begin{array}{l} -1 \\ -1 \\ -3 \\ -4 \end{array}\right)+r\left(\begin{array}{l} -1 \\ -4 \\ -2 \\ -6 \end{array}\right)\)
where \(p, q\) and \(r\) are given real numbers, then
\(\mathbf{x}=\left(\begin{array}{c} p+\lambda \\ q+\lambda \\ r+\lambda \\ \lambda \end{array}\right),\)
where \(\lambda\) is real.

Find the values of \(p, q\) and \(r\) such that
\(p\left(\begin{array}{l} 1 \\ 2 \\ 3 \\ \end{array}\right)+q\left(\begin{array}{l} -1 \\ -1 \\ -3 \\ -4 \end{array}\right)+r\left(\begin{array}{l} -1 \\ -4 \\ -2 \\ -6 \end{array}\right)=\left(\begin{array}{r} 3 \\ 7 \\ 8 \\ \end{array}\right) .\)

Find the solution \(\mathbf{x}=\left(\begin{array}{l}\alpha \\ \beta \\ \gamma \\ \delta\end{array}\right)\) of the equation \(\mathbf{A} \mathbf{x}=\left(\begin{array}{r}3 \\ 7 \\ 8 \\ 15\end{array}\right)\) for which \(\alpha^{2}+\beta^{2}+\gamma^{2}+\delta^{2}=\frac{11}{4}\).

The vector \(\mathbf{e}\) is an eigenvector of the matrix \(\mathbf{A}\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf{B}\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf{e}\) is an eigenvector of the matrix \(\mathbf{A B}\) with corresponding eigenvalue \(\lambda \mu\).

State the eigenvalues of the matrix \(\mathbf{C}\), where
\(\mathbf{C}=\left(\begin{array}{rrr} -1 & -1 & 3 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \end{array}\right),\)
and find corresponding eigenvectors.

Show that \(\left(\begin{array}{l}1 \\ 6 \\ 3\end{array}\right)\) is an eigenvector of the matrix \(\mathbf{D}\), where
\(\mathbf{D}=\left(\begin{array}{rrr} 1 & -1 & 1 \\ -6 & -3 & 4 \\ -9 & -3 & 7 \end{array}\right),\)
and state the corresponding eigenvalue.

Hence state an eigenvector of the matrix \(\mathbf{C D}\) and give the corresponding eigenvalue.

Answer only one of the following two alternatives.
EITHER
The curve \(C\) has equation \(y=\frac{1}{3} x^{\frac{1}{2}}(3-x)\), for \(0 \leqslant x \leqslant 3\). Find the mean value of \(y\) with respect to over the interval \(0 \leqslant x \leqslant 3\).

Show that
\(\frac{\mathrm{d} s}{\mathrm{~d} x}=\frac{1}{2}\left(x^{-\frac{1}{2}}+x^{\frac{1}{2}}\right),\)
where \(s\) denotes arc length, and find the arc length of \(C\).

Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

OR
Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrr} 1 & 1 & 2 \\ 0 & 2 & 2 \\ -1 & 1 & 3 \end{array}\right) .\)

The linear transformation \(\mathrm{T}: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}\) is defined by \(\mathbf{x} \mapsto \mathbf{A x}\). Let \(\mathbf{e}, \mathbf{f}\) be two linearly independent eigenvectors of \(\mathbf{A}\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively, and let \(\Pi\) be the plane, through the origin, containing \(\mathbf{e}\) and \(\mathbf{f}\). By considering the parametric equation of \(\Pi\), show that all points of \(\Pi\) are mapped by T onto points of \(\Pi\).

Find cartesian equations of three planes, each with the property that all points of the plane are mapped by T onto points of the same plane.

The matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & 1 & 4 \\ 1 & 5 & -1 \\ 2 & 1 & 5 \end{array}\right)\)
has eigenvalues \(1,5,7\). Find a set of corresponding eigenvectors.

Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{n}=\mathbf{P D P}^{-1}\).
[The evaluation of \(\mathbf{P}^{-1}\) is not required.]
Determine the set of values of the real constant \(k\) such that \(k^{n} \mathbf{A}^{n}\) tends to the zero matrix as \(n \rightarrow \infty\).

Given that 5 is an eigenvalue of the matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 5 & -3 & 0 \\ 1 & 2 & 1 \\ -1 & 3 & 4 \end{array}\right),\)
find a corresponding eigenvector.

Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf{A}+\mathbf{A}^{2}\).

Answer only one of the following two alternatives.
EITHER

Prove by induction that
\(\sum_{n=1}^{N} n^{3}=\frac{1}{4} N^{2}(N+1)^{2} .\)

Use this result, together with the formula for \(\sum_{n=1}^{N} n^{2}\), to show that
\(\sum_{n=1}^{N}\left(20 n^{3}+36 n^{2}\right)=N(N+1)(N+3)(5 N+2) .\)

Let
\(S_{N}=\sum_{n=1}^{N}\left(20 n^{3}+36 n^{2}+\mu n\right) .\)

Find the value of the constant \(\mu\) such that \(S_{N}\) is of the form \(N^{2}(N+1)(a N+b)\), where the constants \(a\) and \(b\) are to be determined.

Show that, for this value of \(\mu\),
\(5+\frac{22}{N}\lt N^{-4} S_{N}\lt 5+\frac{23}{N}\)
for all \(N \geqslant 18\).

OR
One of the eigenvalues of the matrix
\(\mathbf{A}=\left(\begin{array}{rrr} 1 & -4 & 6 \\ 2 & -4 & 2 \\ -3 & 4 & a \end{array}\right)\)
is -2 . Find the value of \(a\).

Another eigenvalue of \(\mathbf{A}\) is -5 . Find eigenvectors \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) corresponding to the eigenvalues -2 and -5 respectively.

The linear space spanned by \(\mathbf{e}_{1}\) and \(\mathbf{e}_{2}\) is denoted by \(V\).
(i) Prove that, for any vector \(\mathbf{x}\) belonging to \(V\), the vector \(\mathbf{A x}\) also belongs to \(V\).

(ii) Find a non-zero vector which is perpendicular to every vector in \(V\), and determine whether it is an eigenvector of \(\mathbf{A}\).