4.03l Singular/non-singular matrices

It is given that the eigenvalues of the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 4 & 1 & -1 \\ -4 & -1 & 4 \\ 0 & -1 & 5 \end{pmatrix},$$ are \(1, 3, 4\). Find a set of corresponding eigenvectors. [4] Write down a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{M}^n = \mathbf{P}\mathbf{D}^n\mathbf{P}^{-1},$$ where \(n\) is a positive integer. [2] Find \(\mathbf{P}^{-1}\) and deduce that $$\lim_{n \to \infty} 4^{-n}\mathbf{M}^n = \begin{pmatrix} -\frac{1}{3} & 0 & -\frac{1}{3} \\ \frac{4}{3} & 0 & \frac{4}{3} \\ \frac{1}{3} & 0 & \frac{1}{3} \end{pmatrix}.$$ [5]

Find the rank of the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 1 & 1 & 2 & 3 \\ 4 & 3 & 5 & 10 \\ 6 & 6 & 13 & 13 \\ 14 & 12 & 23 & 45 \end{pmatrix}.$$ [3] Find vectors \(\mathbf{x_0}\) and \(\mathbf{e}\) such that any solution of the equation $$\mathbf{A}\mathbf{x} = \begin{pmatrix} 0 \\ 2 \\ -1 \\ -3 \end{pmatrix} \quad (*)$$ can be expressed in the form \(\mathbf{x_0} + \lambda\mathbf{e}\), where \(\lambda \in \mathbb{R}\). [5] Hence show that there is no vector which satisfies \((*)\) and has all its elements positive. [3]

The linear transformation \(\mathrm{T} : \mathbb{R}^4 \to \mathbb{R}^4\) is represented by the matrix \(\mathbf{M}\), where $$\mathbf{M} = \begin{pmatrix} 1 & -2 & -3 & 1 \\ 3 & -5 & -7 & 7 \\ 5 & -9 & -13 & 9 \\ 7 & -13 & -19 & 11 \end{pmatrix}.$$ Find the rank of \(\mathbf{M}\) and a basis for the null space of \(\mathrm{T}\). [6] The vector \(\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \end{pmatrix}\) is denoted by \(\mathbf{e}\). Show that there is a solution of the equation \(\mathbf{M}\mathbf{x} = \mathbf{M}\mathbf{e}\) of the form $$\mathbf{x} = \begin{pmatrix} a \\ b \\ -1 \\ -1 \end{pmatrix}, \text{ where the constants } a \text{ and } b \text{ are to be found.}$$ [4]

The matrix A is given by $$\mathbf{A} = \begin{pmatrix} -6 & 2 & 13 \\ 0 & -2 & 5 \\ 0 & 0 & 8 \end{pmatrix}.$$

1. Find a matrix P and a diagonal matrix D such that \(\mathbf{A}^{-1} = \mathbf{PDP}^{-1}\). [7]
2. Use the characteristic equation of A to find \(\mathbf{A}^{-1}\). [4]

The matrix \(\mathbf{A}\) is given by $$\mathbf{A} = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 7 & 9 \\ 4 & 1 & 7 \end{pmatrix}.$$

1. Show that the characteristic equation of \(\mathbf{A}\) is \(\lambda^3 - 12\lambda^2 + 124 + 80 = 0\) and find the eigenvalues of \(\mathbf{A}\). [4]
2. Use the characteristic equation of \(\mathbf{A}\) to show that $$\mathbf{A}^4 = p\mathbf{A}^2 + q\mathbf{A} + r\mathbf{I},$$ where \(p\), \(q\) and \(r\) are integers to be determined. [4]
3. Find a matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that \((\mathbf{A} - 3\mathbf{I})^4 = \mathbf{PDP}^{-1}\). [6]

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

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]

The transformation \(R\) is represented by the matrix \(\mathbf{A}\), where $$\mathbf{A} = \begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}.$$

1. Find the eigenvectors of \(\mathbf{A}\). [5]
2. Find an orthogonal matrix \(\mathbf{P}\) and a diagonal matrix \(\mathbf{D}\) such that $$\mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}.$$ [5]
3. Hence describe the transformation \(R\) as a combination of geometrical transformations, stating clearly their order. [4]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that $$\mathbf{A} = \begin{bmatrix} 2 & a & 3 \\ 0 & -2 & 1 \end{bmatrix} \quad \text{and} \quad \mathbf{B} = \begin{bmatrix} 1 & -3 \\ -2 & 4a \\ 0 & 5 \end{bmatrix}$$

1. Find the product \(\mathbf{AB}\) in terms of \(a\). [2 marks]
2. Find the determinant of \(\mathbf{AB}\) in terms of \(a\). [1 mark]
3. Show that \(\mathbf{AB}\) is singular when \(a = -1\) [2 marks]

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) are such that \(\mathbf{A} = \begin{bmatrix} 4 & 2 \\ -1 & -3 \end{bmatrix}\) and \(\mathbf{B} = \begin{bmatrix} 4 & 2 \\ 2 & 1 \end{bmatrix}\).

1. Explain why \(\mathbf{B}\) has no inverse. [1]
2. 1. Find the inverse of \(\mathbf{A}\). [3]
   2. Hence, find the matrix \(\mathbf{X}\), where \(\mathbf{AX} = \begin{bmatrix} -4 \\ 1 \end{bmatrix}\) [2]

The matrix \(\mathbf{A}\) is defined by $$\mathbf{A} = \begin{pmatrix} 4 & 8 & 0 \\ 0 & \lambda & -2 \\ 4 & 0 & \lambda \end{pmatrix}.$$

1. Show that there are two values of \(\lambda\) for which \(\mathbf{A}\) is singular. [4]
2. Given that \(\lambda = 3\),
   1. determine the adjugate matrix of \(\mathbf{A}\),
   2. determine the inverse matrix \(\mathbf{A}^{-1}\). [5]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} \lambda & 1 & 14 \\ -1 & 2 & 8 \\ -3 & 2 & \lambda \end{pmatrix}\), where \(\lambda\) is a real constant.

1. Find an expression for the determinant of \(\mathbf{A}\) in terms of \(\lambda\). Give your answer in the form \(a\lambda^2 + b\lambda + c\), where \(a\), \(b\), \(c\) are integers whose values are to be determined. [3]
2. Show that \(\mathbf{A}\) is non-singular for all values of \(\lambda\). [4]

A linear transformation T of the \(x\)-\(y\) plane has an associated matrix M, where \(\mathbf{M} = \begin{pmatrix} \lambda & k \\ 1 & \lambda - k \end{pmatrix}\), and \(\lambda\) and \(k\) are real constants.

1. You are given that \(\det \mathbf{M} > 0\) for all values of \(\lambda\).
   1. Find the range of possible values of \(k\). [3]
   2. What is the significance of the condition \(\det \mathbf{M} > 0\) for the transformation T? [1]

For the remainder of this question, take \(k = -2\).

1. Determine whether there are any lines through the origin that are invariant lines for the transformation T. [4]

$$\mathbf{M} = \begin{pmatrix} 2 & -1 & 1 \\ 3 & k & 4 \\ 3 & 2 & -1 \end{pmatrix} \quad \text{where } k \text{ is a constant}$$

1. Find the values of \(k\) for which the matrix \(\mathbf{M}\) has an inverse. [2]
2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect \begin{align} 2x - y + z &= p
   3x - 6y + 4z &= 1
   3x + 2y - z &= 0 \end{align} [5]
3. 1. Find the value of \(q\) for which the set of simultaneous equations \begin{align} 2x - y + z &= 1
      3x - 5y + 4z &= q
      3x + 2y - z &= 0 \end{align} can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically. [4]

Three matrices, A, B and C, are given by \(\mathbf{A} = \begin{pmatrix} 1 & 2 \\ a & -1 \end{pmatrix}\), \(\mathbf{B} = \begin{pmatrix} 2 & -1 \\ 4 & 1 \end{pmatrix}\) and \(\mathbf{C} = \begin{pmatrix} 5 & 0 \\ -2 & 2 \end{pmatrix}\) where \(a\) is a constant.

1. Using A, B and C in that order demonstrate explicitly the associativity property of matrix multiplication. [4]
2. Use A and C to disprove by counterexample the proposition 'Matrix multiplication is commutative'. [2]

For a certain value of \(a\), \(\mathbf{A}\begin{pmatrix} x \\ y \end{pmatrix} = 3\begin{pmatrix} x \\ y \end{pmatrix}\)

1. Find