4.03b Matrix operations: addition, multiplication, scalar

2 You are given that \(\mathbf { A } = \left( \begin{array} { r } 4 \\ - 2 \\ 4 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r } 5 & 1 \\ 2 & - 3 \end{array} \right) , \mathbf { C } = \left( \begin{array} { l l l } 5 & 1 & 8 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { r r } - 2 & 0 \\ 4 & 1 \end{array} \right)\).

1. Calculate, where they exist, \(\mathbf { A B } , \mathbf { C A } , \mathbf { B } + \mathbf { D }\) and \(\mathbf { A C }\) and indicate any that do not exist.
2. Matrices \(\mathbf { B }\) and \(\mathbf { D }\) represent transformations B and D respectively. Find the single matrix that represents transformation B followed by transformation D.

\(\mathbf { 9 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 2 & 1 & - 5 \\ 3 & a & 1 \\ 1 & - 1 & 2 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { c c c } 2 a + 1 & 3 & 1 + 5 a \\ - 5 & 1 & - 13 \\ - 3 - a & - 1 & - 2 a - 3 \end{array} \right)\).

1. Show that \(\mathbf { A B } = ( 8 + a ) \mathbf { I }\).
2. State the value of \(a\) for which \(\mathbf { A } ^ { - 1 }\) does not exist. Write down \(\mathbf { A } ^ { - 1 }\) in terms of \(a\), when \(\mathbf { A } ^ { - 1 }\) exists.
3. Use \(\mathbf { A } ^ { - 1 }\) to solve the following simultaneous equations. $$\begin{aligned} - 2 x + y - 5 z & = - 55 \\ 3 x + 4 y + z & = - 9 \\ x - y + 2 z & = 26 \end{aligned}$$
4. What can you say about the solutions of the following simultaneous equations? $$\begin{aligned} - 2 x + y - 5 z & = p \\ 3 x - 8 y + z & = q \\ x - y + 2 z & = r \end{aligned}$$

\(\mathbf { 1 }\) You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 2 & - 1 & 1 \\ 0 & p & - 4 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r } 0 & q \\ 2 & - 2 \\ 1 & - 3 \end{array} \right)\).

1. Find \(\mathbf { A B }\).
2. Hence prove that matrix multiplication is not commutative.

9 You are given that \(\mathbf { M } = \left( \begin{array} { l l } 3 & 0 \\ 0 & 2 \end{array} \right) , \mathbf { N } = \left( \begin{array} { l l } 0 & 1 \\ 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { r r } 0 & - 1 \\ 1 & 0 \end{array} \right)\).

1. The matrix products \(\mathbf { Q } ( \mathbf { M N } )\) and \(( \mathbf { Q M } ) \mathbf { N }\) are identical. What property of matrix multiplication does this illustrate? Find QMN. \(\mathbf { M } , \mathbf { N }\) and \(\mathbf { Q }\) represent the transformations \(\mathrm { M } , \mathrm { N }\) and Q respectively.
2. Describe the transformations \(\mathrm { M } , \mathrm { N }\) and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa71f270-53cb-44ba-b3a6-3953fa5c4232-4_668_908_788_621} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure}
3. The points \(\mathrm { A } , \mathrm { B }\) and C in the triangle in Fig. 9 are mapped to the points \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\) respectively by the composite transformation N followed by M followed by Q . Draw a diagram showing the image of the triangle after this composite transformation, labelling the image of each point clearly.

9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0 \\ 0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.

1. Describe fully the transformations P and Q . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
   Triangle T is transformed first by transformation P , then by transformation Q .
2. Find the single matrix that represents this composite transformation.
3. This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). T' is reflected in the line \(y = - x\) to give a new triangle, T".
4. Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
5. A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.

1

1. Write down the matrix for a rotation of \(90 ^ { \circ }\) anticlockwise about the origin.
2. Write down the matrix for a reflection in the line \(y = x\).
3. Find the matrix for the composite transformation of rotation of \(90 ^ { \circ }\) anticlockwise about the origin, followed by a reflection in the line \(y = x\).
4. What single transformation is equivalent to this composite transformation?

1 You are given that the matrix \(\left( \begin{array} { r r } - 1 & 0 \\ 0 & 1 \end{array} \right)\) represents a transformation \(A\), and that the matrix \(\left( \begin{array} { r r } 0 & 1 \\ - 1 & 0 \end{array} \right)\) represents a transformation B .

1. Describe the transformations A and B .
2. Find the matrix representing the composite transformation consisting of A followed by B .
3. What single transformation is represented by this matrix?

9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } - 3 & - 4 & 1 \\ 2 & 1 & k \\ 7 & - 1 & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { r r c } - 4 & - 5 & 11 \\ - 19 & - 4 & - 7 \\ - 9 & - 31 & 2 - k \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } 79 & 0 & - 3 - k \\ - 9 k - 27 & - 31 k - 14 & q \\ p & 0 & 82 + k \end{array} \right)\) where \(p\) and \(q\) are to be determined.

1. Show that \(p = 0\) and \(q = 15 + 2 k - k ^ { 2 }\). It is now given that \(k = - 3\).
2. Find \(\mathbf { A B }\) and hence write down the inverse matrix \(\mathbf { A } ^ { - 1 }\).
3. Use a matrix method to find the values of \(x , y\) and \(z\) that satisfy the equation \(\mathbf { A } \left( \begin{array} { l } x \\ y \\ z \end{array} \right) = \left( \begin{array} { r } 14 \\ - 23 \\ 9 \end{array} \right)\). \section*{THERE ARE NO QUESTIONS WRITTEN ON THIS PAGE}

9 You are given that \(\mathbf { A } = \left( \begin{array} { r r r } 1 & 3 & - 1 \\ - 1 & \alpha & - 1 \\ - 2 & - 1 & 3 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c c } 3 \alpha - 1 & - 8 & \alpha - 3 \\ 5 & 1 & 2 \\ 2 \alpha + 1 & - 5 & \alpha + 3 \end{array} \right)\) and \(\mathbf { A B } = \left( \begin{array} { c c c } \gamma & 0 & 0 \\ \beta & \gamma & 0 \\ 0 & 0 & \gamma \end{array} \right)\).

1. Show that \(\beta = 0\).
2. Find \(\gamma\) in terms of \(\alpha\).
3. Write down \(\mathbf { A } ^ { - 1 }\) for the case when \(\alpha = 2\). State the value of \(\alpha\) for which \(\mathbf { A } ^ { - 1 }\) does not exist.
4. Use your answer to part (iii) to solve the following simultaneous equations. $$\begin{aligned} x + 3 y - z & = 25 \\ - x + 2 y - z & = 11 \\ - 2 x - y + 3 z & = - 23 \end{aligned}$$

3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 4 & - 6 & - 6 \\ 5 & 11 & 10 \end{array} \right)$$ has \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1 \\ - 1 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 2 \\ - 3 \end{array} \right)\), \(\left( \begin{array} { r } 1 \\ 1 \\ - 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
[0pt] [You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]

2 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 1 \end{array} \right)\). Prove by mathematical induction that, for every positive integer \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) \\ 0 & 1 \end{array} \right)$$

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.

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.

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 \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 } \\ x _ { 2 } \\ x _ { 3 } \\ x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]

The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that

1. \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
2. \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\). Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0 \\ 1 & 0 & 1 \\ - 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0 \\ 1 & - 8 & 1 \\ - 1 & 3 & - 6 \end{array} \right)$$