Exercises

After completing the tutorial attempt the following exercises.

If you are not sure how to do something, have a look at the “How To” section.

1. Obtain the determinant and the inverses of the following matrices:

   1. \(A = \begin{pmatrix} 1 / 5 & 1\\1 & 1\end{pmatrix}\)

   2. \(B = \begin{pmatrix} 1 / 5 & 1 & 5\\3 & 1 & 6 \\ 1 & 2 & 1\end{pmatrix}\)

   3. \(C = \begin{pmatrix} 1 / 5 & 5 & 5\\3 & 1 & 7 \\ a & b & c\end{pmatrix}\)

2. Compute the following:

   1. \(500\begin{pmatrix} 1 / 5 & 1\\1 & 1\end{pmatrix}\)

   2. \(\pi \begin{pmatrix} 1 / \pi & 2\pi\\3/\pi & 1\end{pmatrix}\)

   3. \(500\begin{pmatrix} 1 / 5 & 1\\1 & 1\end{pmatrix} + \pi \begin{pmatrix} 1 / \pi & 2\pi\\3/\pi & 1\end{pmatrix}\)

   4. \(500\begin{pmatrix} 1 / 5 & 1\\1 & 1\end{pmatrix}\begin{pmatrix} 1 / \pi & 2\pi\\3/\pi & 1\end{pmatrix}\)

3. The matrix \(A\) is given by \(A=\begin{pmatrix}a & 4 & 2\\ 1 & a & 0\\ 1 & 2 & 1\end{pmatrix}\).

   1. Find the determinant of \(A\)

   2. Hence find the values of \(a\) for which \(A\) is singular.

   3. State, giving a brief reason in each case, whether the simultaneous equations

      \[\begin{split} \begin{array}{l} a x + 4y + 2z= 3a\\ x + a y = 1\\ x + 2y + z = 3\\ \end{array} \end{split}\]

      have any solutions when:

      1. \(a = 3\);

      2. \(a = 2\)

4. The matrix \(D\) is given by \(D = \begin{pmatrix} a & 2 & 0\\ 3 & 1 & 2\\ 0 & -1 & 1\end{pmatrix}\) where \(a\ne 2\).

   1. Find \(D^{-1}\).

   2. Hence or otherwise, solve the equations:

   \[\begin{split} \begin{array}{l} a x + 2y = 3\\ 3x + y + 2z = 4\\ - y + z = 1\\ \end{array} \end{split}\]