Exercises 13 (Matrices and determinants)

  1. The matrices \(A,B\) and \(C\) are given by: \[A=\begin{pmatrix}2&9\\6&1\end{pmatrix},\qquad B=\begin{pmatrix}5&9\\2&4\end{pmatrix},\qquad C=\begin{pmatrix}6&-8\\5&-2\end{pmatrix}.\] Determine:

    1. \(A+C\)
    2. \(B+C\)
    3. \(B-A\)
    4. \(A-C\)
    5. \(C\cdot B\)
    6. \(B\cdot C\)

  2. Calculate:

    1. \(\begin{pmatrix}7&3\\1&6\end{pmatrix}+\begin{pmatrix}2&5\\4&7\end{pmatrix}\)
    2. \(\begin{pmatrix}5&-2\\7&9\end{pmatrix}-\begin{pmatrix}5&-4\\8&-1\end{pmatrix}\)
    3. \(\begin{pmatrix}12&7\\9&3\end{pmatrix}\cdot \begin{pmatrix}6&2\\5&-3\end{pmatrix}\)

  3. For the matrices shown below, determine \(A\cdot B\) and \(B\cdot A\) when possible: \[A=\begin{pmatrix}5&2&9\\3&1&4\\6&2&3\end{pmatrix},\qquad B=\begin{pmatrix}1&2&3\\4&-5&6\end{pmatrix}.\]

  4. Evaluate the following determinants:

    1. \(\begin{vmatrix}2&3\\5&4\end{vmatrix}\)
    2. \(\begin{vmatrix}2&-3\\6&8\end{vmatrix}\)
    3. \(\begin{vmatrix}x&2x\\x^2&-5x\end{vmatrix}\)
    4. \(\begin{vmatrix}1&-5&4\\6&2&8\\1&-3&5\end{vmatrix}\)
    5. \(\begin{vmatrix}3&2&2\\3&-8&2\\3&9&2\end{vmatrix}\)

Solutions: 1. (i) \(\begin{pmatrix}8&1\\11&-1\end{pmatrix}\); (ii) \(\begin{pmatrix}11&1\\7&2\end{pmatrix}\); (iii) \(\begin{pmatrix}3&0\\-4&3\end{pmatrix}\); (iv) \(\begin{pmatrix}-4&17\\1&3\end{pmatrix}\); (v) \(\begin{pmatrix}14&22\\21&37\end{pmatrix}\); (vi) \(\begin{pmatrix}75&-58\\32&-24\end{pmatrix}\); 2. (i) \(\begin{pmatrix}9&8\\5&13\end{pmatrix}\); (ii) \(\begin{pmatrix}0&2\\-1&10\end{pmatrix}\); (iii) \(\begin{pmatrix}107&3\\69&9\end{pmatrix}\); 3. (i) undefined; (ii) \(\begin{pmatrix}29&10&26\\41&15&34\end{pmatrix}\); 4. (i) \(-7\); (ii) \(34\); (iii) \(-5x^2-2x^3\); (iv) \(64\); (v) \(0\);