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 \(\mathbf{A}\).
(a) Find the values of \(a\) for which the system of equations
\(\begin{aligned}\frac{3}{2}x+3y+8z&=1\\ax+3y+4z&=2\\ay-z&=3\end{aligned}\)
does not have a unique solution.
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\begin{pmatrix}\frac{3}{2}&3&8\\0&3&4\\0&0&-1\end{pmatrix}\).
(b) Given that \(\mathbf{B}=\mathbf{A}^{-1}\), use the characteristic equation of \(\mathbf{A}\) to show that \(\mathbf{B}^{2}=p\mathbf{I}+q\mathbf{A}\), where \(p\) and \(q\) are constants to be determined.
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{-1}=\mathbf{P}\mathbf{D}\mathbf{P}^{-1}\).
(a) Find the set of values of \(a\) for which the system of equations
\(\begin{array}{c} 6 x+a y=3 \\ 2 x-y=1 \\ x+5 y+4 z=2 \end{array}\)
has a unique solution.
(b) Show that the system of equations in part (a) is consistent for all values of \(a\).
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 6 & 0 & 0 \\ 2 & -1 & 0 \\ 1 & 5 & 4 \end{array}\right)\)
(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((14 \mathbf{A}+24 \mathbf{I})^{2}=\mathbf{P D P}^{-1}\).
(d) Use the characteristic equation of \(\mathbf{A}\) to show that
\((14 \mathbf{A}+24 \mathbf{I})^{2}=\mathbf{A}^{4}(\mathbf{A}+b \mathbf{I})^{2}\)
where \(b\) is an integer to be determined.
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{array}\right) .\)
(a) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{-1}=\mathbf{P D P} \mathbf{P}^{-1}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 2 & -3 & -7 \\ 0 & 5 & 7 \\ 0 & 0 & -2 \end{array}\right) .\)
(a) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{5}=\mathbf{P D P}^{-1}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to show that
\(\mathbf{A}^{4}=a \mathbf{A}^{2}+b \mathbf{I},\)
where \(a\) and \(b\) are integers to be determined.
The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} -1 & 2 & 12 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{array}\right) .\)
Use the characteristic equation of \(\mathbf{A}\) to show that
\(\mathbf{A}^{4}=p \mathbf{A}^{2}+q \mathbf{I},\)
where \(p\) and \(q\) are integers to be determined.
The matrix \(\mathbf{A}\) is given by
\[\mathbf{A}=\left(\begin{array}{rrr}
5 & -1 & 7 \\
0 & 6 & 0 \\
7 & 7 & 5
\end{array}\right)\]
(a) Find the eigenvalues of \(\mathbf{A}\).
(b) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{-1}\).
The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{M}\), where
\(\mathbf{M}=\left(\begin{array}{rrrr} 1 & -2 & -3 & 1 \\ 3 & -5 & -7 & 7 \\ 5 & -9 & -13 & 9 \\ 7 & -13 & -19 & 11 \end{array}\right) .\)
Find the rank of \(\mathbf{M}\) and a basis for the null space of T .
The vector \(\left(\begin{array}{l}1 \\ 2 \\ 3 \\ 4\end{array}\right)\) is denoted by \(\mathbf{e}\). Show that there is a solution of the equation \(\mathbf{M} \mathbf{x}=\mathbf{M e}\) of the form \(\mathbf{x}=\left(\begin{array}{c}a \\ b \\ -1 \\ -1\end{array}\right)\), where the constants \(a\) and \(b\) are to be found.
EITHER
The linear transformation \(T:\mathbb R^4\to\mathbb R^4\) is represented by
\(\mathbf M=\begin{pmatrix}1&2&3&4\\1&-1&2&3\\1&-3&3&5\\1&4&2&2\end{pmatrix}\).
The range space of \(T\) is denoted by \(V\).
(i) Determine the dimension of \(V\).
(ii) Show that the vectors \(\begin{pmatrix}1\\1\\1\\1\end{pmatrix}\), \(\begin{pmatrix}2\\-1\\-3\\4\end{pmatrix}\), and \(\begin{pmatrix}3\\2\\3\\2\end{pmatrix}\) are a basis of \(V\).
The set of elements of \(\mathbb R^4\) which do not belong to \(V\) is denoted by \(W\).
(iii) State, with a reason, whether \(W\) is a vector space.
(iv) Show that if \(\begin{pmatrix}x\\y\\z\\t\end{pmatrix}\) belongs to \(W\), then \(x+y\ne z+t\).
The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrrr} 1 & 3 & 5 & 7 \\ 2 & 8 & 7 & 9 \\ 3 & 13 & 9 & 11 \\ 6 & 24 & 21 & 27 \end{array}\right)\)
Find
(i) the rank of \(\mathbf{A}\),
(ii) a basis for the range space of T ,
(iii) a basis for the null space of T .
(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 \(\mathbf{A}^{3}\).