4.03h Determinant 2x2: calculation

\(\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix}\), where \(k\) is a constant.

1. Show that the matrix \(\mathbf{A}\) is non-singular for all values of \(k\). [2]

A transformation \(T : \mathbb{R}^2 \to \mathbb{R}^2\) is represented by the matrix \(\mathbf{A}\). The point \(P\) has position vector \(\begin{pmatrix} a \\ 2a \end{pmatrix}\) relative to an origin \(O\). The point \(Q\) has position vector \(\begin{pmatrix} 7 \\ -3 \end{pmatrix}\) relative to \(O\). Given that the point \(P\) is mapped onto the point \(Q\) under \(T\),

1. determine the value of \(a\) and the value of \(k\). [3]

Given that, for a different value of \(k\), \(T\) maps the line \(y = 2x\) onto itself,

1. determine this value of \(k\). [3]

The triangle \(T\) has vertices at the points \((1, k)\), \((3,0)\) and \((11,0)\), where \(k\) is a positive constant. Triangle \(T\) is transformed onto the triangle \(T'\) by the matrix $$\begin{pmatrix} 6 & -2 \\ 1 & 2 \end{pmatrix}$$ Given that the area of triangle \(T'\) is 364 square units, find the value of \(k\). [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]

You are given that \(M = \begin{pmatrix} \frac{1}{\sqrt{3}} & -\sqrt{3} \\ \frac{1}{\sqrt{3}} & 1 \end{pmatrix}\).

1. Show that \(M\) is non-singular. [2]

The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(M\). Given that the area of hexagon \(R\) is 5 square units,

1. find the area of hexagon \(S\). [1]

The matrix \(M\) represents an enlargement, with centre \((0,0)\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \((0,0)\).

1. Find the value of \(k\). [2]
2. Find the value of \(\theta\). [2]

$$\mathbf{A} = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}$$

1. Find \(\mathbf{A}^{-1}\) [2 marks]
2. The determinant of \(\mathbf{A}^2\) is equal to 4. Find the possible values of \(k\). [3 marks]

$$\mathbf{A} = \begin{pmatrix} k & -2 \\ 1-k & k \end{pmatrix},$$ where \(k\) 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]

\(A = \begin{bmatrix} 2 & 3 \\ k & 1 \end{bmatrix}\)

1. Find \(A^{-1}\) [2 marks]
2. The determinant of \(A^2\) is equal to 4. Find the possible values of \(k\). [3 marks]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 0.6 & 2.4 \\ -0.8 & 1.8 \end{pmatrix}\).

1. Find \(\det \mathbf{A}\). [1]

The matrix \(\mathbf{A}\) represents a stretch parallel to one of the coordinate axes followed by a rotation about the origin.

1. By considering the determinants of these transformations, determine the scale factor of the stretch. [2]
2. Explain whether the stretch is parallel to the \(x\)-axis or the \(y\)-axis, justifying your answer. [1]
3. Find the angle of rotation. [2]

The matrix **M** is given by \(\mathbf{M} = \begin{pmatrix} -\frac{3}{5} & \frac{4}{5} \\ \frac{4}{5} & \frac{3}{5} \end{pmatrix}\).

1. The diagram in the Printed Answer Booklet shows the unit square \(OABC\). The image of the unit square under the transformation represented by **M** is \(OA'B'C'\). Draw and clearly label \(OA'B'C'\). [3]
2. Find the equation of the line of invariant points of this transformation. [3]
3. 1. Find the determinant of **M**. [1]
   2. Describe briefly how this value relates to the transformation represented by **M**. [2]

Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]

0 & 2 & 2
- 1 & 1 & 3 \end{array} \right) .$$

1. Find the eigenvalues of \(\mathbf { A }\).
2. Use the characteristic equation of \(\mathbf { A }\) to find \(\mathbf { A } ^ { - 1 }\).

7 The square matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Find the eigenvalues of the matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { l l l }