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10.18 Matrices

Tutorials and Workshops.pdf

  1. Let \(A=\begin{pmatrix}3 & 2 & 4\\ 1 & 0 & 6\end{pmatrix}\in M_{2\times3}(\mathbb{Z}_7),B=\begin{pmatrix}1 & 5\\ 2 & 0\\ 4 & 1\end{pmatrix}\in M_{3\times2}(\mathbb{Z}_7)\) and \(C=\begin{pmatrix}23 & 65\\ 14 & 6\end{pmatrix}\in M_{2\times2}(\mathbb{Z}_7)\)
    Calculate \(A \cdot B + C\)

    Since we are in the same field, thus we can directly calculate and modular in the end.

    Which means that you can also modular and calculate it and modular again if needed.

    Note that \(A\cdot B=\begin{pmatrix}3 & 2 & 4\\ 1 & 0 & 6\end{pmatrix}\cdot\begin{pmatrix}1 & 5\\ 2 & 0\\ 4 & 1\end{pmatrix}=\begin{pmatrix}23 & 19\\ 25 & 11\end{pmatrix}\)​

    \(A\cdot B+C=\begin{pmatrix}23 & 19\\ 25 & 11\end{pmatrix}\begin{pmatrix}23 & 65\\ 14 & 6\end{pmatrix}=\begin{pmatrix}46 & 84\\ 39 & 17\end{pmatrix}=\begin{pmatrix}4 & 0\\ 4 & 3\end{pmatrix}\)

  2. Show that \(A = \begin{pmatrix} 4 & 5 \\ 1 & 6 \end{pmatrix}\) has an inverse in \(M_{2 \times 2}(\mathbb{R})\) and write down the inverse explicitly.

    Since \(ad-bc=24-5=19\neq 0\), then the matrix has an inverse.

    By formular, \(A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}=\frac{1}{19}\begin{pmatrix}6 & -5\\ -1 & 4\end{pmatrix}=\begin{pmatrix}\frac{6}{19} & -\frac{5}{19}\\ -\frac{1}{19} & \frac{4}{19}\end{pmatrix}\)​

  3. Find the values of \(a, b \in \mathbb{R}\) such that the remainder of the polynomial division of \(p(x) = 3x^3 + 4x^2 - 2ax + b\) by \(q(x) = x^2 + x + 1\) is \(r(x) = -4x + 2\).

    Let do the polynomial division.

    \[ \begin{array}{rcl} & &3X + 1 \\ \hline X^2 + X + 1 & \bigg| & 3X^3 + 4X^2 - 2aX + b \\ & & -(3X^3 + 3X^2 + 3X) \\ \hline & & X^2 + X(-2a - 3) + b \\ & & -(X^2 + X + 1) \\ \hline & & X(-2a - 4) + (b - 1) \\ \end{array} \]

    We wish to find \(a, b \in \mathbb{R}\) such that

    \(x(-2a - 4) + (b - 1) = r(x) = -4x + 2\)

    \(\iff x(-2a) + b - 1 - 2 = 0\)

    \(\iff -2ax + b - 3 = 0\)

    \(\iff \begin{cases} -2a = 0 \\ b - 3 = 0 \end{cases} \iff \begin{cases} a = 0 \\ b = 3 \end{cases}\)

    So for \(a = 0\) and \(b = 3\), we have the remainder \(r(x) = -4x + 2\).

  4. Let \(p(x) = x^5 + x^4 + 2x^2 + 1\) and \(q(x) = 2x^3 + x^2 + 1\). Find the remainder of the polynomial division of \(p(x)\) by \(q(x)\) in \(\mathbb{Z}_3[x]\).

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  5. Show that \(p(x) = x^3 + x^2 + x + 1\) is divisible by \(g(x) = x + 3\) in \(\mathbb{Z}_5[x]\) but not in \(\mathbb{Z}[x]\).

    Let's do the division (in \(\mathbb{Z}_5[x]\)).

    image