8 Matrices exercises

This section contains purely bonus exercises to accompany the Matrices chapter. Each chapter contained embedded examples and exercises, often with explanations.

These questions are provided with fewer explanations, generally just with the basic answers provided later. Of course, you can ask questions of the course lecturer about any problems here too.

8.1 Matrix algebra

Consider the following matrices: \[\begin{equation*} A = \begin{pmatrix}1 & 0 & -3 \end{pmatrix}, \quad B=\begin{pmatrix}4 & -1 & 2 \end{pmatrix}, \quad C = \begin{pmatrix}1 & 2 \\ 4 & 3 \end{pmatrix} \end{equation*}\] \[\begin{equation*} D = \begin{pmatrix}-3 & 1 \\ 6 & 5 \end{pmatrix}, \quad E=\begin{pmatrix}-2 & 6 & 3 \\ 1 & 0 & -1 \\ 5 & 8 & 4 \end{pmatrix}, \quad F = \begin{pmatrix}9& 0 & 7 \\ 2 & -2 & 0 \\ 1 & 6 & 5 \end{pmatrix} \end{equation*}\] \[\begin{equation*} G = \begin{pmatrix}2 & 6 \\ 1 & 3 \\ -1 & 5 \end{pmatrix}, \quad H=\begin{pmatrix}0 & -6 \\ -1 & 0 \\ 3 & 8 \end{pmatrix} \end{equation*}\] \[\begin{equation*} J = \begin{pmatrix}2 \\ -1 \\ 4 \end{pmatrix}, \quad K = \begin{pmatrix}-1 \\ 2 \\ 0 \end{pmatrix} \end{equation*}\]
1. State the shape of each matrix.
2. Determine the following algebraic combinations (not all may be possible!)
  1. \(A+B\)
  2. \(C-D\)
  3. \(E+F\)
  4. \(E-F\)
  5. \(2G+3H\)
  6. \(3C-D\)
  7. \(GC\)
  8. \(EJ\)
  9. \(JE\)
  10. \(CD\)
  11. \(DC\)
  12. \(EF\)
  13. \(HE\)
  14. \(AK\)
  15. \(KA\)
Simplify the following linear combinations:
1. \(\begin{pmatrix}1 & 0 \\ 2 & 1 \end{pmatrix}+ \begin{pmatrix}-1 & -2 \\ 2 & 1 \end{pmatrix}\)
2. \(\begin{pmatrix}1 & 3 \\ \frac{1}{2} & -2 \end{pmatrix}+ \begin{pmatrix}-\frac{1}{2} & 0 \\ \frac{3}{2} & 1 \end{pmatrix}\)
3. \(3\begin{pmatrix}1 & -1 \\ 0 & 2 \end{pmatrix}\)
4. \(2\begin{pmatrix}1 & 4 \\ -1 & 6 \end{pmatrix}+ 3\begin{pmatrix}0 & -2 \\ 4 & 1 \end{pmatrix}\)
Calculate the following matrix products:
1. \(\begin{pmatrix}1 & 0 \\ 2 & 1 \end{pmatrix}\begin{pmatrix}-1 & -2 \\ 2 & 1 \end{pmatrix}\)
2. \(\begin{pmatrix}1 & 3 \\ 2 & 1 \end{pmatrix}\begin{pmatrix}1 & 0 & -3 \\ 1 & 2 & -2 \end{pmatrix}\)
3. \(\begin{pmatrix}1 & \frac{1}{2} & 3 \end{pmatrix}\begin{pmatrix}1 & 4 \\ -1 & \frac{1}{2} \\ 0 & 1 \end{pmatrix}\)
4. \(\begin{pmatrix}3 & -2 & 5 \end{pmatrix}\begin{pmatrix}2 \\ 0 \\ 7 \end{pmatrix}\)

8.2 Matrix properties

Calculate the following products:
1. \(\begin{pmatrix}1 & 0 \\ 2 & 1 \end{pmatrix}\left[ \begin{pmatrix}1 & 0 \\ 2&1 \end{pmatrix}\begin{pmatrix}-1 & -2 \\ 2 & 1 \end{pmatrix}\right]\)
2. \(\left[\begin{pmatrix}1 & 0 \\ 2&1 \end{pmatrix}\begin{pmatrix}-1 & -2 \\ 2 & 1 \end{pmatrix}\right]\begin{pmatrix}1 & 0 \\ 2 & 1 \end{pmatrix}\)
3. Look at the two calculations you have just performed, are the answers the same? What does this tell you about the order or matrix multiplication?
Answer the following two questions about squaring matrices:
1. Which of the following matrices can be squared? \(A=\begin{pmatrix}1 & 0 \\ 2& 1 \end{pmatrix}\) and \(M=\begin{pmatrix}1 &0&-3 \\ 1 & 2 & -2\end{pmatrix}\)
2. In general, considering all possible matrices, which matrices can be squared?
For the matrix \(A=\begin{pmatrix}a & b & c \\ d & e & f \end{pmatrix}\),
1. Calculate \(A^T\) and \((A^T)^T\).
2. How does \((A^T)^T\) compare to \(A\)?
Let \(A=\begin{pmatrix}a&b&c \\ d&e&f \end{pmatrix}\) and \(B=\begin{pmatrix}u&v&w \\ x&y&z \end{pmatrix}\).
1. Evaluate \(A^T+B^T\) and \(\left(A+B\right)^T\).
2. Comment on your answer.

8.3 Matrix determinants and inverses

Calculate the determinants of each of the following matrices:
1. \(\begin{pmatrix}1&1 \\ 0 & 1 \end{pmatrix}\)
2. \(\begin{pmatrix}3&-2\\4&5 \end{pmatrix}\)
3. \(\begin{pmatrix}6&-3\\-4&2 \end{pmatrix}\)
4. \(\begin{pmatrix}1&1&0\\0&1&1\\1&0&1 \end{pmatrix}\)
5. \(\begin{pmatrix}1&2&1\\3&2&1\\2&3&2 \end{pmatrix}\)
6. \(\begin{pmatrix}2&1&1\\1&2&2\\2&4&4 \end{pmatrix}\)
For each matrix in the previous question, determine its inverse or explain why you know it doesn’t have an inverse. (You may use a computer for \(3\)-by-\(3\) matrix inverses)
For the matrix \(A=\begin{pmatrix}4 &5 \\ 2 & 3 \end{pmatrix}\), use the standard formula to determine its inverse (written \(A^{-1}\)) and calculate the two products \(AA^{-1}\) and \(A^{-1}A\). Comment on your results.
For the matrix \(A=\begin{pmatrix}4&1&3 \\ 2&1&2 \end{pmatrix}\) calculate the product \(AA^T\). Can you always multiply a matrix by its own transpose?
Let \(A=\begin{pmatrix}2&5\\1&4\end{pmatrix}\) and \(B=\begin{pmatrix}4&-5\\0&3\end{pmatrix}\).
1. First calculate \(A^{-1}\) and \(B^{-1}\)
2. Next evaluate, by hand, \(AB, \left(AB\right)^{-1}\) and \(B^{-1}A^{-1}\)
3. Comment on whether you were expecting these final two evaluations to be equal
Consider the following matrix \(A=\begin{pmatrix}1&2\\ k&3 \end{pmatrix}\) where \(k\) is a constant.
1. Determine which value(s) of \(k\) allow \(A\) to be invertible.
2. Calculate the inverse of \(A\) (note your answer will contain \(k\))
Let \(A=\begin{pmatrix}1&-2&1\\-3&2&-1\\2&-1&0\end{pmatrix}\).
1. Calculate the determinant of \(A\).
2. Find the inverse of \(A\) (using your computer).
3. By hand, calculate the matrix products \(AA^{-1}\) and \(A^{-1}A\).
4. Did your results agree with what you were expecting?
Given that \(D=\begin{pmatrix}2&0&0\\0&5&0\\0&0&4 \end{pmatrix}\), find the inverse matrix \(D^{-1}\) without using a computer.
(Hardest) Let \(A=\begin{pmatrix}4&0&9\\ 0 & 5+k & -3 \\ 0 & 2 & k \end{pmatrix}\) be a matrix, where \(k\) is a constant.
1. Calculate the determinant of \(A\) (your answer will contain \(k\)).
2. Use your determinant formula to determine all values of \(k\) when the matrix is invertible.

8.4 Solving simultaneous of equations

Consider the following system of simultaneous equations: \[\begin{alignat*}{3} 2x & {}-{} & 5y & {}={} & 2 \\ 3x & {}-{} & 7y & {}={} & 1 \end{alignat*}\]
1. Express these simultaneous equations in matrix form, i.e. as \(A\underline{x}=\underline{b}\), where \(A\) is a square matrix, and both \(\underline{b}\) and \(\underline{x}\) are single column matrices.
2. Determine the inverse, \(A^{-1}\), of your matrix \(A\).
3. Use \(A^{-1}\) to find the solution to the simultaneous equations.
A system of equations is given by \[\begin{alignat*}{4} 2x & {}-{} & y & {}+{} & 3z & {}={} & 13 \\ x & {}-{} & 2y & {}-{} & 3z & {}={} & -4 \\ 4x & {}-{} & 2y & {}-{} & 3z & {}={} & 8 \end{alignat*}\]
1. Express these simultaneous equations in the matrix form \[ A\underline{x}=\underline{b}.\]
2. Determine the matrix \(A^{-1}\) (use a computer, it will exist in this case).
3. Use your result in the previous part to solve this system of equations for \(x,y,z\).

7 Vectors exercises

9 Algebra exercises – Solutions