4.03a Matrix language: terminology and notation

A \(3 \times 3\) matrix \(\mathbf { A }\) has eigenvalues \(- 1,1,2\), with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right) , \quad \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) ,$$ respectively. Find

1. the matrix \(\mathbf { A }\),
2. \(\mathbf { A } ^ { 2 n }\), where \(n\) is a positive integer.

Determine the rank of the matrix $$\mathbf { A } = \left( \begin{array} { l l l l } 1 & - 1 & - 1 & 1 \\ 2 & - 1 & - 4 & 3 \\ 3 & - 3 & - 2 & 2 \\ 5 & - 4 & - 6 & 5 \end{array} \right)$$ Show that if $$\mathbf { A x } = p \left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 5 \end{array} \right) + q \left( \begin{array} { l } - 1 \\ - 1 \\ - 3 \\ - 4 \end{array} \right) + r \left( \begin{array} { l } - 1 \\ - 4 \\ - 2 \\ - 6 \end{array} \right)$$ 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 $$p \left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 5 \end{array} \right) + q \left( \begin{array} { l } - 1 \\ - 1 \\ - 3 \\ - 4 \end{array} \right) + r \left( \begin{array} { l } - 1 \\ - 4 \\ - 2 \\ - 6 \end{array} \right) = \left( \begin{array} { r } 3 \\ 7 \\ 8 \\ 15 \end{array} \right) .$$ 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 \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 11 } { 4 }\).

5 A matrix \(\mathbf { A }\) has eigenvalues \(- 1,1\) and 2 , with corresponding eigenvectors $$\left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 3 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 2 \\ - 3 \\ 5 \end{array} \right) ,$$ respectively. Find \(\mathbf { A }\).

7 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & - 3 & 18 & 37 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & - 1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{array} \right)$$ respectively. The null space of \(\mathrm { T } _ { 1 }\) is denoted by \(K _ { 1 }\) and the null space of \(\mathrm { T } _ { 2 }\) is denoted by \(K _ { 2 }\). Show that the dimension of \(K _ { 1 }\) is 2 and that the dimension of \(K _ { 2 }\) is 1 . Find the basis of \(K _ { 1 }\) which has the form \(\left\{ \left( \begin{array} { c } p \\ q \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { c } r \\ s \\ 0 \\ 1 \end{array} \right) \right\}\) and show that \(K _ { 2 }\) is a subspace of \(K _ { 1 }\).

7 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & 4 \\ 2 & 1 & 4 & 11 \\ 3 & 4 & 1 & 9 \\ 4 & - 3 & 18 & 37 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & 1 & 1 & - 1 \\ 2 & 3 & 0 & 1 \\ 3 & 4 & 1 & 0 \\ 4 & 5 & 2 & 0 \end{array} \right)$$ respectively. The null space of \(\mathrm { T } _ { 1 }\) is denoted by \(K _ { 1 }\) and the null space of \(\mathrm { T } _ { 2 }\) is denoted by \(K _ { 2 }\). Show that the dimension of \(K _ { 1 }\) is 2 and that the dimension of \(K _ { 2 }\) is 1 . Find the basis of \(K _ { 1 }\) which has the form \(\left\{ \left( \begin{array} { c } p \\ q \\ 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } r \\ s \\ 0 \\ 1 \end{array} \right) \right\}\) and show that \(K _ { 2 }\) is a subspace of \(K _ { 1 }\).

5 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 $$\mathbf { B } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right) .$$ 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 $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & 2 & - 3 \\ 2 & - 1 & 3 \\ - 3 & 3 & 0 \end{array} \right) ,$$ and state corresponding eigenvectors.

6 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 4 & - 5 & 3 \\ 3 & - 4 & 3 \\ 1 & - 1 & 2 \end{array} \right)$$ 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 $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 4 & 0 \\ - 1 & 3 & 1 \\ 1 & - 1 & 3 \end{array} \right)$$ Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { B }\) and deduce an eigenvector of the matrix \(\mathbf { A B }\), stating the corresponding eigenvalue.

8 The linear transformations \(\mathrm { T } _ { 1 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) and \(\mathrm { T } _ { 2 } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) are represented by the matrices \(\mathbf { M } _ { 1 }\) and \(\mathbf { M } _ { 2 }\) respectively, where $$\mathbf { M } _ { 1 } = \left( \begin{array} { r r r r } 1 & - 2 & 3 & 5 \\ 3 & - 4 & 17 & 33 \\ 5 & - 9 & 20 & 36 \\ 4 & - 7 & 16 & 29 \end{array} \right) \quad \text { and } \quad \mathbf { M } _ { 2 } = \left( \begin{array} { r r r r } 1 & - 2 & 0 & - 3 \\ 2 & - 1 & 0 & 0 \\ 4 & - 7 & 1 & - 9 \\ 6 & - 10 & 0 & - 14 \end{array} \right) .$$ The null spaces of \(\mathrm { T } _ { 1 }\) and \(\mathrm { T } _ { 2 }\) are denoted by \(K _ { 1 }\) and \(K _ { 2 }\) respectively. Find a basis for \(K _ { 1 }\) and a basis for \(K _ { 2 }\). It is given that \(\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\). The vectors \(\mathbf { x } _ { 1 }\) and \(\mathbf { x } _ { 2 }\) are such that \(\mathbf { M } _ { 1 } \mathbf { x } _ { 1 } = \mathbf { M } _ { 1 } \mathbf { a }\) and \(\mathbf { M } _ { 2 } \mathbf { x } _ { 2 } = \mathbf { M } _ { 2 } \mathbf { a }\). Given that \(\mathbf { x } _ { 1 } - \mathbf { x } _ { 2 } = \left( \begin{array} { c } p \\ 5 \\ 7 \\ q \end{array} \right)\), find \(p\) and \(q\).

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } - 2 & 5 & 3 & - 1 \\ 0 & 1 & - 4 & - 2 \\ 6 & - 14 & - 13 & 1 \\ \alpha & \alpha & - 2 \alpha & - 11 \alpha \end{array} \right)$$ and \(\alpha\) is a constant. The null space of T is denoted by \(K _ { 1 }\) when \(\alpha \neq 0\), and by \(K _ { 2 }\) when \(\alpha = 0\). Find a basis for \(K _ { 1 }\) and a basis for \(K _ { 2 }\).

9 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 $$\mathbf { A } = \left( \begin{array} { r r r } - 1 & 2 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{array} \right)$$ Write down the eigenvalues of \(\mathbf { A }\) and obtain corresponding eigenvectors. Given that $$\mathbf { M } = \left( \begin{array} { l l l } 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right)$$ find the eigenvalues and corresponding eigenvectors of \(\mathbf { B }\).

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & - 1 & 1 & 3 \\ 2 & 0 & 0 & 5 \\ 6 & - 2 & 2 & 11 \\ 10 & - 3 & 3 & 19 \end{array} \right)$$

1. Find the rank of \(\mathbf { M }\) and state a basis for the range space of T .
2. Obtain a basis for the null space of T .

9 The matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } - 2 & 2 & 2 \\ 2 & 1 & 2 \\ - 3 & - 6 & - 7 \end{array} \right)$$ 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 $$\mathbf { A } = \left( \begin{array} { l l l } a & b & c \\ d & e & f \\ g & h & i \end{array} \right)$$ 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.

8 It is given that \(\lambda\) is an eigenvalue of the non-singular square matrix \(\mathbf { A }\), with corresponding eigenvector
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 $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ Find a corresponding eigenvector. 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 the matrix \(\mathbf { A }\). Find the corresponding eigenvalues.
Hence find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\left( \mathbf { A } + \mathbf { A } ^ { - 1 } \right) ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }$$

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 2 & - 3 \\ 2 & 2 & 3 \\ - 3 & 3 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) has an eigenvector \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\). Find the corresponding eigenvalue. The matrix \(\mathbf { A }\) also has eigenvalues 4 and 6. Find corresponding eigenvectors. Hence find a matrix \(\mathbf { P }\) such that \(\mathbf { A } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\), where \(\mathbf { D }\) is a diagonal matrix which is to be determined. The matrix \(\mathbf { B }\) is such that \(\mathbf { B } = \mathbf { Q A Q } ^ { - 1 }\), where $$\mathbf { Q } = \left( \begin{array} { r r r } 4 & 11 & 5 \\ 1 & 4 & 2 \\ 1 & 2 & 1 \end{array} \right)$$ By using the expression \(\mathbf { P D P } ^ { - 1 }\) for \(\mathbf { A }\), find the set of eigenvalues and a corresponding set of eigenvectors for \(\mathbf { B }\).
[0pt] [Question 11 is printed on the next page.]

10 Write down the eigenvalues of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 1 \\ 0 & - 1 & 2 \\ 0 & 0 & 1 \end{array} \right)$$ 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.
[0pt] [Question 11 is printed on the next page.]

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & 3 & - 4 \\ 2 & - 4 & 7 & - 9 \\ 4 & - 8 & 14 & - 18 \\ 5 & - 10 & 17 & - 22 \end{array} \right)$$ Find the rank of \(\mathbf { M }\). Obtain a basis for the null space \(K\) of T . Evaluate $$\mathbf { M } \left( \begin{array} { r } 1 \\ - 2 \\ 2 \\ - 1 \end{array} \right)$$ and hence show that any solution of $$\mathbf { M x } = \left( \begin{array} { l } 15 \\ 33 \\ 66 \\ 81 \end{array} \right)$$

5 The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & - 2 \\ 6 & 4 & - 6 \\ 6 & 5 & - 7 \end{array} \right)$$ has eigenvalues \(1 , - 1\) and - 2 .

1. Find a set of corresponding eigenvectors.
2. 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.

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 8 & 7 \\ 7 & - 9 & 7 \\ 6 & - 6 & 5 \end{array} \right)$$

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

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

1. Find the rank of \(\mathbf { A }\) and show that \(\left\{ \left( \begin{array} { r } 2 \\ 2 \\ - 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 1 \\ 3 \\ 0 \\ 1 \end{array} \right) \right\}\) is a basis for the null space of the transformation. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Show that if $$\mathbf { A x } = p \left( \begin{array} { r } 1 \\ 3 \\ 5 \\ - 2 \end{array} \right) + q \left( \begin{array} { r } - 1 \\ - 1 \\ - 8 \\ 3 \end{array} \right) ,$$ where \(p\) and \(q\) are given real numbers, then $$\mathbf { x } = \left( \begin{array} { c } p + 2 \lambda + \mu \\ q + 2 \lambda + 3 \mu \\ - \lambda \\ \mu \end{array} \right) ,$$ where \(\lambda\) and \(\mu\) are real numbers. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Find the values of \(p\) and \(q\) such that $$p \left( \begin{array} { r } 1 \\ 3 \\ 5 \\ - 2 \end{array} \right) + q \left( \begin{array} { r } - 1 \\ - 1 \\ - 8 \\ 3 \end{array} \right) = \left( \begin{array} { r } 3 \\ 7 \\ 18 \\ - 7 \end{array} \right)$$ \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
4. Find the solution of the equation \(\mathbf { A x } = \left( \begin{array} { r } 3 \\ 7 \\ 18 \\ - 7 \end{array} \right)\) of the form \(\mathbf { x } = \left( \begin{array} { l } 4 \\ 9 \\ \alpha \\ \beta \end{array} \right)\), where \(\alpha\) and \(\beta\) are positive integers to be found. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)

8 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r c r } 1 & 2 & \alpha & - 1 \\ 2 & 6 & - 3 & - 3 \\ 3 & 10 & - 6 & - 5 \end{array} \right)$$ and \(\alpha\) is a constant. When \(\alpha \neq 0\) the null space of T is denoted by \(K _ { 1 }\).

1. Find a basis for \(K _ { 1 }\).
   When \(\alpha = 0\) the null space of T is denoted by \(K _ { 2 }\).
2. Find a basis for \(K _ { 2 }\).
3. Determine, justifying your answer, whether \(K _ { 1 }\) is a subspace of \(K _ { 2 }\).

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

1. Write down another eigenvector of \(\mathbf { A }\) corresponding to \(\lambda\).
2. Write down an eigenvector and corresponding eigenvalue of \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
   Let \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0 \\ 2 & 7 & 0 \\ 4 & 8 & 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
4. Determine the set of values of the real constant \(k\) such that $$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

5 It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\) with corresponding eigenvalue \(\lambda\).

1. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 3 }\) and state the corresponding eigenvalue.
   It is given that $$\mathbf { A } = \left( \begin{array} { r r } 2 & 0 \\ - 1 & 3 \end{array} \right) .$$
2. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { A } ^ { 3 } + \mathbf { I } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.

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

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

A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) , \quad \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) , \quad \left( \begin{array} { r } 0 \\ 1 \\ - 1 \end{array} \right)$$ respectively, where \(b\) is a positive constant.

1. Find \(\mathbf { A }\) in terms of \(b\).
2. Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0 \\ 2 \\ - 2 \end{array} \right)\).
3. It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right) = \left( \begin{array} { l } 4 \\ 4 \\ 0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1 \\ 0 \\ b \end{array} \right) = \left( \begin{array} { c } - 1 \\ 0 \\ b ^ { - 1 } \end{array} \right) .$$ Find the values of \(n\) and \(b\).

10 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { H }\), where $$\mathbf { H } = \left( \begin{array} { r r r r } 1 & 2 & - 3 & - 5 \\ - 1 & 4 & 5 & 1 \\ 2 & 3 & 0 & - 3 \\ - 3 & 5 & 7 & 2 \end{array} \right)$$

1. Find the dimension of the range space of T .
2. Find a basis for the null space of \(T\).
3. It is given that \(\mathbf { x }\) satisfies the equation $$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right)$$ Using the fact that $$\mathbf { H } \left( \begin{array} { r } 1 \\ - 3 \\ 1 \\ - 2 \end{array} \right) = \left( \begin{array} { r } 2 \\ - 10 \\ - 1 \\ - 15 \end{array} \right) ,$$ find the least possible value of \(| \mathbf { x } |\).
   [0pt] [For the vector \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]