Question_2

Question 2

We can only add matrices of the same size. The pairs of matrices with the same size are (A,C) and (D,F). Noting that multiplication by a scalar doesn’t change the size, the sums A - 2C, C - 2A, D - 2F, F - 2D are all defined. We have (taking it relatively slowly):

\begin{align*}A - 2C &= \begin{pmatrix}1 & 2 \\ 3 & 4\end{pmatrix} - 2 \begin{pmatrix}-1 & 5 \\ 4 & 4\end{pmatrix} \\ &= \begin{pmatrix}1&2\\3&4\end{pmatrix} + \begin{pmatrix}2 & -10 \\ -8 & -8\end{pmatrix} \\ & = \begin{pmatrix}1 + 2 & 2 - 10 \\ 3 - 8 & 4 - 8\end{pmatrix} \\ &= \begin{pmatrix}3 & -8 \\ -5 & -4\end{pmatrix}\end{align*}

\begin{align*}C - 2A &= \begin{pmatrix}-1 & 5 \\ 4 & 4\end{pmatrix} - 2 \begin{pmatrix}1&2\\3&4\end{pmatrix} \\ &= \begin{pmatrix}-1&5\\4&4\end{pmatrix} + \begin{pmatrix}-2&-4\\-6&-8\end{pmatrix} \\ &= \begin{pmatrix}-1 - 2 & 5 - 4 \\ 4 - 6 & 4 - 8\end{pmatrix} \\ &= \begin{pmatrix}-3 & 1 \\ -2 & -4\end{pmatrix}\end{align*}

\begin{align*}D - 2F &= \begin{pmatrix}1 & 4 & -3 \\ 2 & 4 & -2\end{pmatrix} - 2 \begin{pmatrix}-1 & 5 & -6 \\ 3 & 4 & - 1\end{pmatrix} \\&= \begin{pmatrix}1&4&-3\\2&4&-2\end{pmatrix} + \begin{pmatrix}2&-10&12\\-6&-8&2\end{pmatrix} \\ &= \begin{pmatrix}1 + 2 & 4 - 10 & -3 + 12\\ 2 - 6 & 4 - 8 & - 2 + 2\end{pmatrix} \\&= \begin{pmatrix}3&-6&9\\-4&-4&0\end{pmatrix}\end{align*}

\begin{align*}F - 2D & = \begin{pmatrix}-1 & 5 & -6 \\ 3 & 4 & -1\end{pmatrix} - 2 \begin{pmatrix}1&4&-3\\2&4&-2\end{pmatrix} \\ & = \begin{pmatrix} -1 & 5 & -6 \\ 3 & 4 & - 1\end{pmatrix} + \begin{pmatrix}-2 & -8 & 6 \\ -4 & -8 & 4\end{pmatrix} \\ & = \begin{pmatrix}-1 - 2 & 5 - 8 & -6 + 6 \\ 3 - 4 & 4 - 8 & -1 + 4\end{pmatrix} \\ & = \begin{pmatrix}-3 & -3 & 0 \\ -1 & -4 & 3\end{pmatrix}\end{align*}

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