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Let $A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 3 & 4\end{array}\right], B=\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]$ and $C=\left[\begin{array}{cccc}2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3\end{array}\right]$, what is $A^T B$ ?
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Verified Answer
The correct answer is:
$\left[\begin{array}{ccc}4 & 0 & -3 \\ -7 & -6 & -6 \\ 4 & -8 & -18\end{array}\right]$
$A=\left[\begin{array}{ccc}1 & -1 & 2 \\ 0 & 3 & 4\end{array}\right], B=\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]$ and $\quad C=\left[\begin{array}{cccc}2 & -3 & 0 & 1 \\ 5 & -1 & -4 & 2 \\ -1 & 0 & 0 & 3\end{array}\right]$
Now,
$$
A^T=\left[\begin{array}{cc}
1 & 0 \\
-1 & 3 \\
2 & 4
\end{array}\right], B=\left[\begin{array}{ccc}
4 & 0 & -3 \\
-1 & -2 & -3
\end{array}\right]
$$
$\Rightarrow A^T B=\left[\begin{array}{cc}1 & 0 \\ -1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]$
$=\left[\begin{array}{ccc}4+0 & 0+0 & -3+0 \\ -4-3 & 0-6 & 3-9 \\ 8-4 & 0-8 & -6-12\end{array}\right]$
$=\left[\begin{array}{ccc}4 & 0 & -3 \\ -7 & -6 & -6 \\ 4 & -8 & -18\end{array}\right]$
Now,
$$
A^T=\left[\begin{array}{cc}
1 & 0 \\
-1 & 3 \\
2 & 4
\end{array}\right], B=\left[\begin{array}{ccc}
4 & 0 & -3 \\
-1 & -2 & -3
\end{array}\right]
$$
$\Rightarrow A^T B=\left[\begin{array}{cc}1 & 0 \\ -1 & 3 \\ 2 & 4\end{array}\right]\left[\begin{array}{ccc}4 & 0 & -3 \\ -1 & -2 & -3\end{array}\right]$
$=\left[\begin{array}{ccc}4+0 & 0+0 & -3+0 \\ -4-3 & 0-6 & 3-9 \\ 8-4 & 0-8 & -6-12\end{array}\right]$
$=\left[\begin{array}{ccc}4 & 0 & -3 \\ -7 & -6 & -6 \\ 4 & -8 & -18\end{array}\right]$
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