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

Show that \(\mathbf{e}\) is an eigenvector of \(\mathbf{A}^{3}\) with corresponding eigenvalue \(\lambda^{3}\).

The matrix \(\mathbf{A}\) is given by

\(\mathbf{A}=\begin{pmatrix}-1&3&4\\0&1&0\\0&-2&5\end{pmatrix}\).

(b) Show that the eigenvalues of \(\mathbf{A}\) are \(-1\), \(1\) and \(5\).

(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}-2\mathbf{I}=\mathbf{P}\mathbf{D}\mathbf{P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to show that \((\mathbf{A}-2\mathbf{I})^{3}=a\mathbf{A}^{2}+b\mathbf{A}+c\mathbf{I}\), where \(a\), \(b\) and \(c\) are constants to be determined.

The planes \(\Pi_{1}\) and \(\Pi_{2}\) do not intersect and are both perpendicular to \(\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}\). The line \(l\) intersects \(\Pi_{1}\) at the point ( \(1,6,0\) ) and intersects \(\Pi_{2}\) at the point ( \(3,-6,0\) ).
(a) Find Cartesian equations of \(\Pi_{1}\) and \(\Pi_{2}\).

(b) Express the vector equation of \(l\) in the form \(\left(\begin{array}{l}x \\ y \\ z\end{array}\right)=\mathbf{a}+\lambda \mathbf{b}\), where \(\mathbf{a}\) and \(\mathbf{b}\) are vectors to be determined, and hence show that for points on \(l, \frac{1}{2} x+\frac{1}{12} y=1\) and \(z=0\).

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{ccc} 1 & 2 & 3 \\ 1 & 2 & 3 \\ \frac{1}{2} & \frac{1}{12} & 0 \end{array}\right) .\)
(c) Show that the characteristic equation of \(\mathbf{A}\) is \(-\lambda^{3}+3 \lambda^{2}+\frac{7}{4} \lambda=0\) and hence find the eigenvalues of \(\mathbf{A}\).
(d) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{n}=\mathbf{P D P}^{-1}\), where \(n\) is a positive integer.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 18 & 5 & -11 \\ 8 & 6 & -4 \\ 32 & 10 & -20 \end{array}\right)\)
(a) Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^{3}-4 \lambda^{2}-20 \lambda+48=0\) and hence find the eigenvalues of \(\mathbf{A}\).
(b) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{array}\right) .\)
(a) Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^{3}-12 \lambda^{2}+12 \lambda+80=0\) and find the eigenvalues of A.

(b) Use the characteristic equation of \(\mathbf{A}\) to show that
\(\mathbf{A}^{4}=p \mathbf{A}^{2}+q \mathbf{A}+r \mathbf{I},\)
where \(p, q\) and \(r\) are integers to be determined.
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A}-3 \mathbf{I})^{4}=\mathbf{P D P}^{-1}\).

(a) 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.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 2 & 0 & 3 \\ 15 & -4 & 3 \\ 3 & 0 & 2 \end{array}\right)\)
(b) Given that -1 is an eigenvalue of \(\mathbf{A}\), find a corresponding eigenvector.
(c) It is also given that \(\left(\begin{array}{l}0 \\ 1 \\ 0\end{array}\right)\) and \(\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right)\) are eigenvectors of \(\mathbf{A}\). Find the corresponding eigenvalues.
(d) Hence find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{-1}=\mathbf{P D P}^{-1}\).
(e) 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 rational numbers to be determined.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 5 & -\frac{22}{3} & 8 \\ 0 & -6 & 0 \\ 0 & 0 & 1 \end{array}\right)\)
(a) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{2}=\mathbf{P D P}^{-1}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{3}\).

(a) Find the values of \(a\) for which the system of equations
\[\begin{array}{r}
3 x+y+z=0 \\
a x+6 y-z=0 \\
a y-2 z=0
\end{array}\]
does not have a unique solution.

The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{rrr}
3 & 1 & 1 \\
0 & 6 & -1 \\
0 & 0 & -2
\end{array}\right)\]
(b) Use the characteristic equation of \(\mathbf{A}\) to find the inverse of \(\mathbf{A}^{2}\).
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).

(a) Find the value of \(a\) for which the system of equations
\[\begin{array}{r}
13 x+18 y-28 z=0 \\
-4 x-a y+8 z=0 \\
2 x+6 y-5 z=0
\end{array}\]
does not have a unique solution.

The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{rrr}
13 & 18 & -28 \\
-4 & -1 & 8 \\
2 & 6 & -5
\end{array}\right)\]
(b) Find the eigenvalue of \(\mathbf{A}\) corresponding to the eigenvector \(\left(\begin{array}{l}2 \\ 0 \\ 1\end{array}\right)\).
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}=\mathbf{P D P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\) in terms of \(\mathbf{A}\).

It is given that \(a\) is a positive constant.
(a) Show that the system of equations
\[\begin{aligned}
a x+(2 a+5) y+(a+1) z & =1 \\
-4 y & =2 \\
3 y-z & =3
\end{aligned}\]
has a unique solution and interpret this situation geometrically.

The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{ccc}
a & 2 a+5 & a+1 \\
0 & -4 & 0 \\
0 & 3 & -1
\end{array}\right)\]
(b) Show that the eigenvalues of \(\mathbf{A}\) are \(a,-1\) and -4 .

(c) Find a matrix \(\mathbf{P}\) such that
\[\mathbf{A}=\mathbf{P}\left(\begin{array}{rrr}
a & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -4
\end{array}\right) \mathbf{P}^{-1}\]
(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).

The matrix \(\mathbf{P}\) is given by
\[\mathbf{P}=\left(\begin{array}{rrr}
1 & 6 & 6 \\
0 & 2 & 6 \\
0 & 0 & -3
\end{array}\right) .\]
(a) Use the characteristic equation of \(\mathbf{P}\) to find \(\mathbf{P}^{-1}\).

(b) Find the matrix \(\mathbf{A}\) such that
\[\mathbf{P}^{-1} \mathbf{A} \mathbf{P}=\left(\begin{array}{ccc}
4 & 0 & 0 \\
0 & 5 & 0 \\
0 & 0 & 6
\end{array}\right) .\]
(c) State the eigenvalues and corresponding eigenvectors of \(\mathbf{A}^{3}\).

The matrix \(\mathbf A\) is given by

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

The matrix \(\mathbf A\) has an eigenvector \(\begin{pmatrix}1\\-1\\1\end{pmatrix}\). Find the corresponding eigenvalue.

The matrix \(\mathbf A\) also has eigenvalues \(4\) and \(6\). Find corresponding eigenvectors.

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

The matrix \(\mathbf B\) is such that \(\mathbf B=\mathbf Q\mathbf A\mathbf Q^{-1}\), where \(\mathbf Q=\begin{pmatrix}4&11&5\\1&4&2\\1&2&1\end{pmatrix}\).

By using the expression \(\mathbf P\mathbf D\mathbf P^{-1}\) for \(\mathbf A\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf B\).

EITHER

It is given that \(1\) and \(4\) are eigenvalues of the matrix \(\mathbf A\), where

\(\mathbf A=\begin{pmatrix}1&-3&-3\\-8&6&-3\\8&-2&7\end{pmatrix}.\)

Find eigenvectors corresponding to each of these eigenvalues.

Given further that \(\begin{pmatrix}0\\1\\-1\end{pmatrix}\) is an eigenvector of \(\mathbf A\), find the corresponding eigenvalue.

Write down matrices \(\mathbf P\) and \(\mathbf D\) such that \(\mathbf P^{-1}\mathbf A\mathbf P=\mathbf D\), where \(\mathbf D\) is a diagonal matrix, and find \(\mathbf P^{-1}\).

Write down a matrix \(\mathbf C\) such that \(\mathbf C^2=\mathbf D\), and deduce a matrix \(\mathbf B\) such that \(\mathbf B^2=\mathbf A\).

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

Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{P}^{-1} \mathbf{A P}=\mathbf{D}\), and hence find the matrix \(\mathbf{A}^{n}\), where \(n\) is a positive integer.
[Question 11 is printed on the next page.]