Matrices

(a)Obtain the matrix, $\displaystyle A$, associated with an anticlockwise rotation of $\displaystyle \frac{\pi}{3}$ radians about the origin.

(b)Find the matrix, $\displaystyle B$, associated with a reflection in the x-axis.

(c)Hence obtain the matrix, $\displaystyle P$, associated with an anticlockwise rotation of $\displaystyle \frac{\pi}{3}$ radians about the origin followed by reflection in the x-axis, expressing your answer using exact values.

(d)Explain why matrix $\displaystyle P$ is not associated with rotation about the origin.

Matrices $\displaystyle A$ and $\displaystyle B$ are defined by $\displaystyle A = \begin{pmatrix} -3 & 2 \\\ 0 & 1 \end{pmatrix}$ and $\displaystyle B = \begin{pmatrix} 2 & 2 \\\ 5 & \lambda \end{pmatrix}$ where $\displaystyle \lambda \in \mathbb{R}.$

(a)Find $\displaystyle 3A + 2B.$

(b)(i) Find $\displaystyle A'B$, where $\displaystyle A'$ is the transpose of $\displaystyle A.$

(b)(ii) Find an expression for the determinant of $\displaystyle A'B.$

(b)(iii) Determine the value of $\displaystyle \lambda$ such that $\displaystyle A'B$ is singular.

The matrix $\displaystyle A$ has the following property:

$\displaystyle A^2 = 6A - I$, where $\displaystyle I$ is the identity matrix.

(a)Express $\displaystyle A^3$ in the form $\displaystyle pA + qI$, where $\displaystyle p, q \in \mathbb{R}.$

Matrix $\displaystyle A$ is non-singular.

(b)Find a similar expression for $\displaystyle A^{-1}$ in terms of $\displaystyle A$ and $\displaystyle I.$

Matrix $\displaystyle A$ is defined by $\displaystyle A=\begin{pmatrix}3&5\\-2&x\end{pmatrix}.$

(a)State an expression for the determinant of $\displaystyle A$ in terms of $\displaystyle x.$

Matrix $\displaystyle A$ is multiplied by matrix $\displaystyle B$ such that $\displaystyle \det AB=12x+40.$

(b)State the determinant of $\displaystyle B.$

The inverse of matrix $\displaystyle B$ is $\displaystyle B^{-1}=\begin{pmatrix}1&-1\\[7pt]-\dfrac{5}{4}&\dfrac{3}{2}\end{pmatrix}.$

(c)Find matrix $\displaystyle B.$