4.03a Matrix language: terminology and notation

6 The matrix A, where $$\mathbf { A } = \left( \begin{array} { r r r } 1 & 0 & 0 \\ 10 & - 7 & 10 \\ 7 & - 5 & 8 \end{array} \right)$$ has eigenvalues 1 and 3. Find corresponding eigenvectors. It is given that \(\left( \begin{array} { l } 0 \\ 2 \\ 1 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\). Find the corresponding eigenvalue. Find a diagonal matrix \(\mathbf { D }\) and matrices \(\mathbf { P }\) and \(\mathbf { P } ^ { - 1 }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\).

7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & - 2 & - 3 & 1 \\ 3 & - 5 & - 7 & 7 \\ 5 & - 9 & - 13 & 9 \\ 7 & - 13 & - 19 & 11 \end{array} \right)$$ Find the rank of \(\mathbf { M }\) and a basis for the null space of T . The vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \\ 4 \end{array} \right)\) is denoted by \(\mathbf { e }\). Show that there is a solution of the equation \(\mathbf { M x } = \mathbf { M e }\) of the form \(\mathbf { x } = \left( \begin{array} { c } a \\ b \\ - 1 \\ - 1 \end{array} \right)\), where the constants \(a\) and \(b\) are to be found.

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 \(\mathbf { c }\).

5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.

1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
   The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1 \\ - 1 & 2 & 3 \\ 1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) as eigenvectors.
2. Find the corresponding eigenvalues.
   The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1 \\ 2 \\ 1 \end{array} \right) , \left( \begin{array} { r } 1 \\ 4 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) respectively.
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).

Three \(n \times 1\) column vectors are denoted by \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\), and \(\mathbf{M}\) is an \(n \times n\) matrix. Show that if \(\mathbf{x}_1\), \(\mathbf{x}_2\), \(\mathbf{x}_3\) are linearly dependent then the vectors \(\mathbf{Mx}_1\), \(\mathbf{Mx}_2\), \(\mathbf{Mx}_3\) are also linearly dependent. [2] The vectors \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) and the matrix \(\mathbf{P}\) are defined as follows: $$\mathbf{y}_1 = \begin{pmatrix} 1 \\ 5 \\ 7 \end{pmatrix}, \quad \mathbf{y}_2 = \begin{pmatrix} 2 \\ -3 \\ 4 \end{pmatrix}, \quad \mathbf{y}_3 = \begin{pmatrix} 5 \\ 51 \\ 55 \end{pmatrix},$$ $$\mathbf{P} = \begin{pmatrix} 1 & -4 & 3 \\ 0 & 2 & 5 \\ 0 & 0 & -7 \end{pmatrix}$$

1. Show that \(\mathbf{y}_1\), \(\mathbf{y}_2\), \(\mathbf{y}_3\) are linearly dependent. [2]
2. Find a basis for the linear space spanned by the vectors \(\mathbf{Py}_1\), \(\mathbf{Py}_2\), \(\mathbf{Py}_3\). [2]

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}, \text{ where } k \text{ is constant.}$$ A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\).

1. Find the value of \(k\) for which the line \(y = 2x\) is mapped onto itself under \(T\). [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(k\). [3]
3. Find \(\mathbf{A}^{-1}\) in terms of \(k\). [2]

A point \(P\) is mapped onto a point \(Q\) under \(T\). The point \(Q\) has position vector \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\) relative to an origin \(O\). Given that \(k = 3\),

1. find the position vector of \(P\). [3]

$$\mathbf{M} = \begin{pmatrix} 1 & k & 0 \\ -1 & 1 & 1 \\ 1 & k & 3 \end{pmatrix}, \text{ where } k \text{ is a constant}$$

1. Find \(\mathbf{M}^{-1}\) in terms of \(k\). [5]

Hence, given that \(k = 0\)

1. find the matrix \(\mathbf{N}\) such that $$\mathbf{MN} = \begin{pmatrix} 3 & 5 & 6 \\ 4 & -1 & 1 \\ 3 & 2 & -3 \end{pmatrix}$$ [4]

$$\mathbf{M} = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 4 & 1 \\ 0 & 5 & 0 \end{pmatrix}$$

1. Show that matrix \(\mathbf{M}\) is not orthogonal. [2]
2. Using algebra, show that \(1\) is an eigenvalue of \(\mathbf{M}\) and find the other two eigenvalues of \(\mathbf{M}\). [5]
3. Find an eigenvector of \(\mathbf{M}\) which corresponds to the eigenvalue \(1\) [2]

The transformation \(M : \mathbb{R}^3 \to \mathbb{R}^3\) is represented by the matrix \(\mathbf{M}\).

1. Find a cartesian equation of the image, under this transformation, of the line $$x = \frac{y}{2} = \frac{z}{-1}$$ [4]

$$\mathbf{A} = \begin{pmatrix} 1 & 0 & 4 \\ 0 & 5 & 4 \\ 4 & 4 & 3 \end{pmatrix}.$$

1. Verify that \(\begin{pmatrix} 2 \\ -2 \\ 1 \end{pmatrix}\) is an eigenvector of \(\mathbf{A}\) and find the corresponding eigenvalue. [3]
2. Show that \(9\) is another eigenvalue of \(\mathbf{A}\) and find the corresponding eigenvector. [5]
3. Given that the third eigenvector of \(\mathbf{A}\) is \(\begin{pmatrix} 2 \\ 1 \\ -2 \end{pmatrix}\), write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{D}.$$ [5]

The matrix \(\mathbf{M}\) is given by $$\mathbf{M} = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 0 & p \\ a & b & c \end{pmatrix},$$ where \(p\), \(a\), \(b\) and \(c\) are constants and \(a > 0\). Given that \(\mathbf{M}\mathbf{M}^T = k\mathbf{I}\) for some constant \(k\), find

1. the value of \(p\), [2]
2. the value of \(k\), [2]
3. the values of \(a\), \(b\) and \(c\), [6]
4. \(|\det \mathbf{M}|\). [2]

\includegraphics{figure_6} The diagram shows the unit square \(OABC\), and its image \(OA'B'C'\) after a transformation. The points have the following coordinates: \(A(1, 0)\), \(B(1, 1)\), \(C(0, 1)\), \(B'(3, 2)\) and \(C'(2, 2)\).

1. Write down the matrix, X, for this transformation. [2]
2. The transformation represented by X is equivalent to a transformation P followed by a transformation Q. Give geometrical descriptions of a pair of possible transformations P and Q and state the matrices that represent them. [6]
3. Find the matrix that represents transformation Q followed by transformation P. [2]

A set of matrices \(M\) is defined by $$A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} \omega & 0 \\ 0 & \omega^2 \end{pmatrix}, \quad C = \begin{pmatrix} \omega^2 & 0 \\ 0 & \omega \end{pmatrix}, \quad D = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad E = \begin{pmatrix} 0 & \omega^2 \\ \omega & 0 \end{pmatrix}, \quad F = \begin{pmatrix} 0 & \omega \\ \omega^2 & 0 \end{pmatrix},$$ where \(\omega\) and \(\omega^2\) are the complex cube roots of 1. It is given that \(M\) is a group under matrix multiplication.

1. Write down the elements of a subgroup of order 2. [1]
2. Explain why there is no element \(X\) of the group, other than \(A\), which satisfies the equation \(X^2 = A\). [2]
3. By finding \(BE\) and \(EB\), verify the closure property for the pair of elements \(B\) and \(E\). [4]
4. Find the inverses of \(B\) and \(E\). [3]
5. Determine whether the group \(M\) is isomorphic to the group \(N\) which is defined as the set of numbers \(\{1, 2, 4, 8, 7, 5\}\) under multiplication modulo 9. Justify your answer clearly. [3]

Two matrices \(\mathbf{A}\) and \(\mathbf{B}\) satisfy the equation $$\mathbf{AB} = \mathbf{I} + 2\mathbf{A}$$ where \(\mathbf{I}\) is the identity matrix and \(\mathbf{B} = \begin{pmatrix} 3 & -2 \\ -4 & 8 \end{pmatrix}\) Find \(\mathbf{A}\). [3 marks]

Which of the following matrices is an identity matrix? Circle your answer. [1 mark] \(\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\) \quad \(\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\)

The set \(S\) is given by \(S = \{0, 2, 4, 6\}\)

1. Construct a Cayley table, using the grid below, for \(S\) under the binary operation addition modulo 8 [3 marks] \includegraphics{figure_2}
2. State the identity element for \(S\) under the binary operation addition modulo 8 [1 mark]

The matrix M is given by $$\mathbf{M} = \frac{1}{10} \begin{pmatrix} a & a & -6 \\ 0 & 10 & 0 \\ 9 & 14 & -13 \end{pmatrix}$$ where \(a\) is a real number. The vectors \(\mathbf{v}_1\), \(\mathbf{v}_2\), and \(\mathbf{v}_3\) are eigenvectors of \(\mathbf{M}\) The corresponding eigenvalues are \(\lambda_1\), \(\lambda_2\), and \(\lambda_3\) respectively. It is given that \(\lambda_2 = 1\) and \(\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}\) and \(\mathbf{v}_3 = \begin{pmatrix} c \\ 0 \\ 1 \end{pmatrix}\), where \(c\) is an integer.

1. 1. Find the value of \(\lambda_1\) [2 marks]
   2. Find the value of \(a\) [2 marks]
2. Find the integer \(c\) and the value of \(\lambda_3\) [4 marks]
3. Find matrices \(\mathbf{U}\), \(\mathbf{D}\) and \(\mathbf{U}^{-1}\), such that \(\mathbf{D}\) is diagonal and \(\mathbf{M} = \mathbf{UDU}^{-1}\) [3 marks]

Josh and Zoe are solving the following mathematics problem: The curve \(C_1\) has equation $$\frac{x^2}{16} - \frac{y^2}{9} = 1$$ The matrix \(\mathbf{M} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\) maps \(C_1\) onto \(C_2\) Find the equations of the asymptotes of \(C_2\) Josh says that to solve this problem you must first carry out the transformation on \(C_1\) to find \(C_2\), and then find the asymptotes of \(C_2\) Zoe says that you will get the same answer if you first find the asymptotes of \(C_1\), and then carry out the transformation on these asymptotes to obtain the asymptotes of \(C_2\) Show that Zoe is correct. [5 marks]

You are given that matrix \(\mathbf{M} = \begin{pmatrix} -3 & 8 \\ -2 & 5 \end{pmatrix}\).

1. Prove that, for all positive integers \(n\), \(\mathbf{M}^n = \begin{pmatrix} 1-4n & 8n \\ -2n & 1+4n \end{pmatrix}\). [6]
2. Determine the equation of the line of invariant points of the transformation represented by the matrix \(\mathbf{M}\). [3]

It is claimed that the answer to part (ii) is also a line of invariant points of the transformation represented by the matrix \(\mathbf{M}^n\), for any positive integer \(n\).

1. Explain geometrically why this claim is true. [2]
2. Verify algebraically that this claim is true. [3]

The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\begin{pmatrix} x & -y \\ y & x \end{pmatrix}\), where \(x\) and \(y\) are real numbers which are not both zero.

1. 1. The matrices \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\) and \(\begin{pmatrix} c & -d \\ d & c \end{pmatrix}\) are both elements of \(X\). Show that \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\begin{pmatrix} c & -d \\ d & c \end{pmatrix} = \begin{pmatrix} p & -q \\ q & p \end{pmatrix}\) for some real numbers \(p\) and \(q\) to be found in terms of \(a\), \(b\), \(c\) and \(d\). [2]
   2. Prove by contradiction that \(p\) and \(q\) are not both zero. [5]
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.] [4]
3. Determine a subgroup of \(G\) of order 17. [2]

Let \(\mathbf{A}\) be the matrix \(\begin{pmatrix} 17 & 12 \\ 12 & 10 \end{pmatrix}\).

1. 1. Determine the integer \(n\) for which \(27\mathbf{A} - \mathbf{A}^2 = n\mathbf{I}\), where \(\mathbf{I}\) is the \(2 \times 2\) identity matrix. [2]
   2. Hence find \(\mathbf{A}^{-1}\) in the form \(p\mathbf{A} + q\mathbf{I}\) for rational numbers \(p\) and \(q\). [2]
2. The plane transformation \(T\) is defined by \(T: \begin{pmatrix} x \\ y \end{pmatrix} \mapsto \mathbf{A} \begin{pmatrix} x \\ y \end{pmatrix}\). It is given that \(T\) is a stretch, with scale factor \(k\), parallel to the line \(y = mx\), where \(m > 0\).
   1. Find the value of \(k\). [2]
   2. By considering \(\mathbf{A} \begin{pmatrix} x \\ mx \end{pmatrix}\), or otherwise, determine the value of \(m\). [4]