Exercise Set 8

  1. Write down the \(2\times 2\) matrix \(R_\pi\) that rotates a vector anticlockwise by \(\pi\). Apply this to a vector \(\mathbf{v}=\left(\begin{smallmatrix}v_1\\v_2\end{smallmatrix}\right)\).

  2. Write down the \(2\times 2\) matrix \(M_x\) that reflects a vector in the \(x\)-axis. Similarly write down the \(2\times 2\) matrix \(M_y\) that reflects a vector in the \(y\)-axis. Multiply these two matrices to find the transformation that first reflects in the \(x\)-axis and then reflects in the \(y\)-axis. Compare this to the matrix \(R_\pi\).

  3. Note that \(R^2_\theta=R_{2\theta}\) (why?). Use this to find the “double angle identity” for \(\cos\) and \(\sin\). Can you find other trigonometric identities using \(R_\theta\)?

  4. For each of the following pairs of matrices \(A\) and \(B\), find (when possible), \(A+B\), \(A-B\), \(A^2\), \(B^2\), \(AB\), \(BA\).

    1. \(A = \begin{pmatrix} 2 & 3 \\ 4 & 1 \end{pmatrix}\qquad B = \begin{pmatrix} -1 & 2 \\ -2 & 0 \end{pmatrix}\)
    2. \(A = \begin{pmatrix} 4 & -5 \\ 6 & 1 \\ 0 & 1 \end{pmatrix}\qquad B = \begin{pmatrix} 5 & 2 & -3 \\ 1 & 3 & -1 \\ 2 & 2 & -1 \end{pmatrix}\)
    3. \(A = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 0 & 1\\ 1 & 0 & 1 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{pmatrix}\)
    4. \(A = \begin{pmatrix} -1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1\\ 1 & 0 & 1 & 0\\ 1 & 1 & 1 & -1 \end{pmatrix} \qquad B = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \\ 1 & -1 & 1 \end{pmatrix}\).

  5. Find two \(3 \times 3\) matrices \(A\) and \(B\) such that \(AB=BA\). Now find two \(3 \times 3\) matrices such that \(AB\neq BA\).

  6. For any two numbers \(a\) and \(b\), if \(ab=0\) then at least one of \(a\) or \(b\) must be \(0\). Does an analagous result hold for matrices? That is, if \(AB=0_{n\times m}\) must at least one of the matrices \(A\) or \(B\) be the zero matrix?

  7. Determine whether the following matrices are invertible or singular by computing their determinants. If they are invertible, find the inverse.

    1. \(\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}\)
    2. \(\begin{pmatrix} 6 & 3 \\ -4 & -2 \end{pmatrix}\)
    3. \(\begin{pmatrix} 4 & -28 & 48 \\ -27 & 162 & -216 \\ 32 & -160 & 192 \end{pmatrix}\)
    4. \(\begin{pmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)\end{pmatrix}\)
    5. \(\begin{pmatrix} 1 & 3 & -5 \\ -2 & 1 & 4 \\ 1 & 2 & -4 \end{pmatrix}\)
    6. \(\begin{pmatrix} 1 & -1 & 4 \\ 2 & 3 & 3 \\ 3 & 1 & 8 \end{pmatrix}\)

  8. Find the eigenvalues and eigenvectors of each of the following matrices \(A\). Determine whether the matrix is diagonalisable and, if so, find the matrices \(D\) and \(P\) in the diagonalisation \(D=P^{-1}AP\).

    1. \(\begin{pmatrix} 1 & 0 \\ 2 & 2 \end{pmatrix}\)
    2. \(\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\)
    3. \(\begin{pmatrix} 1 & 2 \\ 2 & -2 \end{pmatrix}\)
    4. \(\begin{pmatrix} 1 & -2 & -1 \\ 2 & 6 & 2 \\ -1 & -2 & 1 \end{pmatrix}\)
    5. \(\begin{pmatrix} -2 & 1 & 1 \\ -11 & 4 & 5 \\ -1 & 1 & 0 \end{pmatrix}\)
    6. \(\begin{pmatrix} 2 & \sqrt 2 & 0 \\ \sqrt 2 & 2 & \sqrt 2 \\ 0 & \sqrt 2 & 2 \end{pmatrix}\)
    7. \(\begin{pmatrix} 1 & -1 & -1 \\ 1 & -1 & 0 \\ 1 & 0 & -1 \end{pmatrix}\)
    8. \(\begin{pmatrix} 5 & 5 & 1 \\ -2 & -1 & 0 \\ 1 & 1 & 1 \end{pmatrix}\).

  9. For the matrices in a. and d. in the previous question, find a formula for \(A^n\).

  10. Linear Difference Equations. A population of Wildebeest can be classified into two life stages: juvenile and adult. Each year \(60\%\) of the juveniles survive to become adults, adults give birth on average to \(0.5\) juvelines and \(70\%\) of adults survive the year. If there are \(200\) juveniles and \(200\) adults in one year, what is the long term population of juveniles and adults? What is the long term ratio of juveniles to adults? Hint: write this as a matrix equation and use diagonalisation (save some time and use a computer to find the eigenvalues and eigenvectors for this question).

    How about in the case when the adult survival rate increases to \(80\%\)? In this case also give the long term growth rate of the juvenile and adult populations. What happens in the case that the adult survival rate drops to \(60\%\)?