Eigenvalues and eigenvectors

5. $$\mathbf { A } = \left( \begin{array} { r r r } a & a & 1 \\ - a & 4 & 0 \\ 4 & a & 5 \end{array} \right) \quad \text { where } a \text { is a positive constant }$$

1. Determine the exact value of \(a\) for which the matrix \(\mathbf { A }\) is singular. Given that 2 is an eigenvalue of \(\mathbf { A }\)
2. determine
   1. the value of \(a\)
   2. the other two eigenvalues of \(\mathbf { A }\) A normalised eigenvector for the eigenvalue 2 is \(\left( \begin{array} { c } \frac { 1 } { \sqrt { 6 } } \\ \frac { 1 } { \sqrt { 6 } } \\ - \frac { 2 } { \sqrt { 6 } } \end{array} \right)\)
3. Determine a normalised eigenvector for each of the other eigenvalues of \(\mathbf { A }\)

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1. \(\mathbf { M } = \left( \begin{array} { r r r } 0 & 1 & 9 \\ 1 & 4 & k \\ 1 & 0 & - 3 \end{array} \right)\), where \(k\) is a constant.

Given that \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\) is an eigenvector of the matrix \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { r } 7 \\ 19 \\ 1 \end{array} \right)\),
2. show that \(k = - 7\)
3. find the other two eigenvalues of the matrix \(\mathbf { M }\). The image of the vector \(\left( \begin{array} { c } p \\ q \\ r \end{array} \right)\) under the transformation represented by \(\mathbf { M }\) is \(\left( \begin{array} { r } - 6 \\ 21 \\ 5 \end{array} \right)\).
4. Find the values of the constants \(p , q\) and \(r\).

5. $$\mathbf { M } = \left( \begin{array} { r r r } 1 & 2 & k \\ - 1 & - 3 & 4 \\ 2 & 6 & - 8 \end{array} \right) \quad \text { where } k \text { is a constant }$$ Given that \(\mathbf { M }\) has a repeated eigenvalue, determine

1. the possible values of \(k\),
2. all corresponding eigenvalues of \(\mathbf { M }\) for each value of \(k\).

3. $$\mathbf { M } = \left( \begin{array} { r r r } 6 & 1 & - 1 \\ 0 & 7 & 0 \\ 3 & - 1 & 2 \end{array} \right)$$

1. Show that 7 is an eigenvalue of the matrix \(\mathbf { M }\) and find the other two eigenvalues of \(\mathbf { M }\).
2. Find an eigenvector corresponding to the eigenvalue 7.

3. \(\mathbf { M } = \left( \begin{array} { r r r } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\),

1. find the value of \(k\).
2. Hence find the other two eigenvalues of \(\mathbf { M }\).
3. Find an eigenvector corresponding to the eigenvalue 3
   3. \(\quad \mathbf { M } = \left( \begin{array} { r c c } 3 & k & 2 \\ - 1 & 0 & 1 \\ 1 & k & 1 \end{array} \right)\), where \(k\) is a constant Given that 3 is an eigenvalue of \(\mathbf { M }\), (a) find the value of \(k\).

3 This question concerns the matrix \(\mathbf { M }\) where \(\mathbf { M } = \left( \begin{array} { r r r } 5 & - 1 & 3 \\ 4 & - 3 & - 2 \\ 2 & 1 & 4 \end{array} \right)\).

1. Obtain the characteristic equation of \(\mathbf { M }\). Find the eigenvalues of \(\mathbf { M }\). These eigenvalues are denoted by \(\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 }\), where \(\lambda _ { 1 } < \lambda _ { 2 } < \lambda _ { 3 }\).
2. Verify that an eigenvector corresponding to \(\lambda _ { 1 }\) is \(\left( \begin{array} { r } 1 \\ 3 \\ - 1 \end{array} \right)\) and that an eigenvector corresponding to \(\lambda _ { 2 }\) is \(\left( \begin{array} { r } 1 \\ 2 \\ - 1 \end{array} \right)\). Find an eigenvector of the form \(\left( \begin{array} { l } a \\ 1 \\ c \end{array} \right)\) corresponding to \(\lambda _ { 3 }\).
3. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } = \mathbf { P D P } ^ { - 1 }\). (You are not required to calculate \(\mathbf { P } ^ { - 1 }\).) Hence write down an expression for \(\mathbf { M } ^ { 4 }\) in terms of \(\mathbf { P }\) and a diagonal matrix. You should give the elements of the diagonal matrix explicitly.
4. Use the Cayley-Hamilton theorem to obtain an expression for \(\mathbf { M } ^ { 4 }\) as a linear combination of \(\mathbf { M }\) and \(\mathbf { M } ^ { 2 }\).

$$\mathbf{M} = \begin{pmatrix} 4 & -5 \\ 6 & -9 \end{pmatrix}$$

1. Find the eigenvalues of \(\mathbf{M}\). [4]

A transformation \(T: \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{M}\). There is a line through the origin for which every point on the line is mapped onto itself under \(T\).

1. Find a cartesian equation of this line. [3]

$$\mathbf{A} = \begin{pmatrix} 3 & 2 & 4 \\ 2 & 0 & 2 \\ 4 & 2 & k \end{pmatrix}.$$

1. Show that \(\det \mathbf{A} = 20 - 4k\). [2]
2. Find \(\mathbf{A}^{-1}\). [6]

Given that \(k = 3\) and that \(\begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\),

1. find the corresponding eigenvalue. [2]

Given that the only other distinct eigenvalue of \(\mathbf{A}\) is \(8\),

1. find a corresponding eigenvector. [4]

In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.

1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
2. Find a vector equation for the axis of the rotation. [4]
3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]