3x3 Matrices

10 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} x + 4 y + 12 z & = 5 \\ 2 x + a y + 12 z & = a - 1 \\ 3 x + 12 y + 2 a z & = 10 \end{aligned}$$ has a unique solution. Show that the system does not have any solution in the case \(a = 18\). Given that \(a = 8\), show that the number of solutions is infinite and find the solution for which \(x + y + z = 1\).

3 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 3 & - 2 & 4 \\ 5 & 15 & - 9 & 19 \\ - 2 & - 6 & 3 & - 7 \\ 3 & 9 & - 5 & 11 \end{array} \right)\).

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

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 $$\mathbf { M } = \left( \begin{array} { r r r r } 2 & - 1 & 1 & 3 \\ 2 & 0 & 0 & 5 \\ 6 & - 2 & 2 & 11 \\ 10 & - 3 & 3 & 19 \end{array} \right)$$

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 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 \(K _ { 2 }\).

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

1. For \(a \neq - 4\), the range space of T is denoted by \(V\).
   (a) Find the dimension of \(V\) and show that $$\left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right) , \quad \left( \begin{array} { r } 2 \\ 0 \\ - 2 \\ 2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4 \\ - 1 \\ a \\ 2 \end{array} \right)$$ form a basis for \(V\).
   (b) Show that if \(\left( \begin{array} { l } x \\ y \\ z \\ t \end{array} \right)\) belongs to \(V\) then \(x + 2 y = t\).
2. For \(a = - 4\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 1 \\ 1 \\ 1 \end{array} \right)$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 4 & 2 & 5 \\ 6 & 7 & 5 & 8 \\ 9 & 9 & 9 & 9 \\ 15 & 16 & 14 & 17 \end{array} \right)$$ Find

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

5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 3 & 5 & 7 \\ 2 & 8 & 7 & 9 \\ 3 & 13 & 9 & 11 \\ 6 & 24 & 21 & 27 \end{array} \right)$$ Find

1. the rank of \(\mathbf { A }\),
2. a basis for the range space of T ,
3. a basis for the null space of T .

1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2 \\ 9 \\ 0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3 \\ 3 \\ 4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0 \\ - 8 \\ 3 \end{array} \right) .$$

1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).

10 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 5 & 1 \\ 1 & - 2 & - 2 \\ 2 & 3 & \theta \end{array} \right)$$

1. (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 $$\begin{aligned} x + 5 y + z & = - 1 \\ x - 2 y - 2 z & = 0 \\ 2 x + 3 y + \theta z & = \theta \end{aligned}$$
2. Solve the system of equations when \(\theta \neq - 1\).
3. Find the general solution when \(\theta = - 1\).
4. Show that if \(\theta = - 1\) and \(\phi \neq - 1\) then \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)\) has no solution.

10 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 5 & 1 \\ 1 & - 2 & - 2 \\ 2 & 3 & \theta \end{array} \right)$$

1. (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 $$\begin{aligned} x + 5 y + z & = - 1 \\ x - 2 y - 2 z & = 0 \\ 2 x + 3 y + \theta z & = \theta \end{aligned}$$
2. Solve the system of equations when \(\theta \neq - 1\).
3. Find the general solution when \(\theta = - 1\).
4. Show that if \(\theta = - 1\) and \(\phi \neq - 1\) then \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } - 1 \\ 0 \\ \phi \end{array} \right)\) has no solution.

2

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