The matrix \(\mathbf{A}\) is defined by
\(\mathbf{A}=\left(\begin{array}{rrr}
1 & 5 & 1 \\
1 & -2 & -2 \\
2 & 3 & \theta
\end{array}\right) .\)
(i) (a) Find the rank of \(\mathbf{A}\) when \(\theta \neq-1\).

(b) Find the rank of \(\mathbf{A}\) when \(\theta=-1\).

Consider the system of equations
\(\begin{aligned}
x+5 y+z & =-1 \\
x-2 y-2 z & =0 \\
2 x+3 y+\theta z & =\theta
\end{aligned}\)
(ii) Solve the system of equations when \(\theta \neq-1\).

(iii) Find the general solution when \(\theta=-1\).

(iv) Show that if \(\theta=-1\) and \(\phi \neq-1\) then \(\mathbf{A x}=\left(\begin{array}{r}-1 \\ 0 \\ \phi\end{array}\right)\) has no solution.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{ccc} a & -6 a & 2 a+2 \\ 0 & 1-a & 0 \\ 0 & 2-a & -1 \end{array}\right)\)
where \(a\) is a constant with \(a \neq 0\) and \(a \neq 1\).
(a) Show that the equation \(\mathbf{A}\left(\begin{array}{c}x \\ y \\ z\end{array}\right)=\left(\begin{array}{l}1 \\ 2 \\ 3\end{array}\right)\) has a unique solution and interpret this situation geometrically.

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

(c) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{4}=\mathbf{P D P}^{-1}\).

(d) Use the characteristic equation of \(\mathbf{A}\) to find \(\mathbf{A}^{4}\) in terms of \(\mathbf{A}\) and \(a\).

(a) Show that the system of equations
\(\begin{array}{r} x+2 y+3 z=1 \\ 4 x+5 y+6 z=1 \\ 7 x+8 y+9 z=1 \end{array}\)
does not have a unique solution.
(b) Show that the system of equations in part (a) is consistent. Interpret this situation geometrically.

Find the set of values of \(k\) for which the system of equations
\(\begin{array}{r} x+5 y+6 z=1 \\ k x+2 y+2 z=2 \\ -3 x+4 y+8 z=3 \end{array}\)
has a unique solution and interpret this situation geometrically.

Show that the system of equations
\(\begin{aligned} 14 x-4 y+6 z & =5 \\ x+y+k z & =3 \\ -21 x+6 y-9 z & =14 \end{aligned}\)
where \(k\) is a constant, does not have a unique solution and interpret this situation geometrically.

(a) Find the value of \(a\) for which the system of equations
\(\begin{array}{c} 3 x+a y=0 \\ 5 x-y=0 \\ x+3 y+2 z=0 \end{array}\)
does not have a unique solution.

The matrix \(\mathbf{A}\) is given by
\(\mathbf{A}=\left(\begin{array}{rrr} 3 & 0 & 0 \\ 5 & -1 & 0 \\ 1 & 3 & 2 \end{array}\right)\)
(b) Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \(\mathbf{A}^{2}=\mathbf{P D P}^{-1}\).

(c) Use the characteristic equation of \(\mathbf{A}\) to show that
\((\mathbf{A}+6 \mathbf{I})^{2}=\mathbf{A}^{4}(\mathbf{A}+b \mathbf{I})^{2}\)
where \(b\) is an integer to be determined.

(a) Show that the system of equations
\(\begin{array}{l} x-y+2 z=4 \\ x-y-3 z=a \\ x-y+7 z=13 \end{array}\)
where \(a\) is a constant, does not have a unique solution.
(b) Given that \(a=-5\), show that the system of equations in part (a) is consistent. Interpret this situation geometrically.
(c) Given instead that \(a \neq-5\), show that the system of equations in part (a) is inconsistent. Interpret this situation geometrically.

(a) Find the set of values of \(k\) for which the system of equations
\(\begin{aligned} x+2 y+3 z & =1 \\ k x+4 y+6 z & =0 \\ 7 x+8 y+9 z & =3 \end{aligned}\)
has a unique solution.

(a) Given that \(a\) is an integer, show that the system of equations
\(\begin{aligned} a x+3 y+z & =14 \\ 2 x+y+3 z & =0 \\ -x+2 y-5 z & =17 \end{aligned}\)
has a unique solution and interpret this situation geometrically.
(b) Find the value of \(a\) for which \(x=1, y=4, z=-2\) is the solution to the system of equations in part (a).

The matrix \(A\), given by \(A=\begin{pmatrix}1&-1&0&2\\3&-1&4&0\\5&-8&-6&19\\-2&3&2&-7\end{pmatrix}\), represents a transformation from \(\mathbb{R}^4\) to \(\mathbb{R}^4\).

(i) Find the rank of \(A\) and show that \(\left\{\begin{pmatrix}2\\2\\-1\\0\end{pmatrix},\begin{pmatrix}1\\3\\0\\1\end{pmatrix}\right\}\) is a basis for the null space of the transformation.

(ii) Show that if \(A\mathbf{x}=p\begin{pmatrix}1\\3\\5\\-2\end{pmatrix}+q\begin{pmatrix}-1\\-1\\-8\\3\end{pmatrix}\), where \(p\) and \(q\) are given real numbers, then \(\mathbf{x}=\begin{pmatrix}p+2\lambda+\mu\\q+2\lambda+3\mu\\-\lambda\\\mu\end{pmatrix}\), where \(\lambda\) and \(\mu\) are real numbers.

(iii) Find the values of \(p\) and \(q\) such that \(p\begin{pmatrix}1\\3\\5\\-2\end{pmatrix}+q\begin{pmatrix}-1\\-1\\-8\\3\end{pmatrix}=\begin{pmatrix}3\\7\\18\\-7\end{pmatrix}\).

(iv) Find the solution of \(A\mathbf{x}=\begin{pmatrix}3\\7\\18\\-7\end{pmatrix}\) of the form \(\mathbf{x}=\begin{pmatrix}4\\9\\m\\n\end{pmatrix}\), where \(m\) and \(n\) are positive integers to be found.

Find the value of the constant \(k\) for which the system of equations
\(\begin{aligned} 2 x-3 y+4 z & =1 \\ 3 x-y & =2 \\ x+2 y+k z & =1 \end{aligned}\)
does not have a unique solution.

For this value of \(k\), solve the system of equations.

Find the two values of the constant \(k\) for which the equations

\(\begin{aligned}kx+y+z&=2\\x+ky+z&=-1\\x+y+kz&=-1\end{aligned}\)

have no unique solution.

Show that, for one of these values of \(k\), the equations have no solution, and solve the equations for the other value of \(k\).

OR

The linear transformation \(T:\mathbb R^4\to\mathbb R^4\) is represented by the matrix

\(\mathbf M=\begin{pmatrix}1&-2&3&-4\\2&-4&7&-9\\4&-8&14&-18\\5&-10&17&-22\end{pmatrix}.\)

Find the rank of \(\mathbf M\).

Obtain a basis for the null space \(K\) of \(T\).

Evaluate

\(\mathbf M\begin{pmatrix}1\\-2\\2\\-1\end{pmatrix},\)

and hence show that any solution of

\(\mathbf M\mathbf x=\begin{pmatrix}15\\33\\66\\81\end{pmatrix}\)

has the form

\(\begin{pmatrix}1\\-2\\2\\-1\end{pmatrix}+\lambda\mathbf e_1+\mu\mathbf e_2,\)

where \(\lambda\) and \(\mu\) are scalars and \(\{\mathbf e_1,\mathbf e_2\}\) is a basis for \(K\). Hence obtain a solution \(\mathbf x'\) for which the sum of the components is \(6\) and the sum of the squares of the components is \(26\).

The linear transformation \(\mathrm{T}: \mathbb{R}^{4} \rightarrow \mathbb{R}^{4}\) is represented by the matrix \(\mathbf{A}\), where
\(\mathbf{A}=\left(\begin{array}{rrrr} 1 & -1 & -2 & 3 \\ 5 & -3 & -4 & 25 \\ 6 & -4 & -6 & 28 \\ 7 & -5 & -8 & 31 \end{array}\right) .\)
(i) Find the rank of \(\mathbf{A}\) and a basis for the null space of T .

(ii) Find the matrix product \(\mathbf{A}\left(\begin{array}{r}-1 \\ 1 \\ -1 \\ 1\end{array}\right)\) and hence find the general solution of the equation \(\mathbf{A} \mathbf{x}=\left(\begin{array}{r}3 \\ 21 \\ 24 \\ 27\end{array}\right)\).

Show that the matrix \(\left(\begin{array}{rrr}1 & 0 & -2 \\ 3 & -3 & -4\end{array}\right)\) has no inverse.

Solve the system of equations
\(\begin{array}{r} x+4 y+2 z=0 \\ 3 x-2 z=4 \\ 3 x-3 y-4 z=5 \end{array}\)