Question
If $A=\left[\begin{array}{cc}-1 & -5 \\ 2 & 0 \\ 3 & -4\end{array}\right]_{3 \times 2} \quad$ Find $\left(A^T\right)^T$.
✓
Answer
$
\begin{aligned}
& \text { Let } A=\left[\begin{array}{cc}
-1 & -5 \\
2 & 0 \\
3 & -4
\end{array}\right]_{3 \times 2} \\
& \therefore A^{\mathrm{T}}=\left[\begin{array}{ccc}
-1 & 2 & 3 \\
-5 & 0 & -4
\end{array}\right]_{2 \times 3} \\
& \therefore\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=\left[\begin{array}{cc}
-1 & -5 \\
2 & 0 \\
3 & -4
\end{array}\right]_{3 \times 2}=\mathrm{A}
\end{aligned}
$
\begin{aligned}
& \text { Let } A=\left[\begin{array}{cc}
-1 & -5 \\
2 & 0 \\
3 & -4
\end{array}\right]_{3 \times 2} \\
& \therefore A^{\mathrm{T}}=\left[\begin{array}{ccc}
-1 & 2 & 3 \\
-5 & 0 & -4
\end{array}\right]_{2 \times 3} \\
& \therefore\left(A^{\mathrm{T}}\right)^{\mathrm{T}}=\left[\begin{array}{cc}
-1 & -5 \\
2 & 0 \\
3 & -4
\end{array}\right]_{3 \times 2}=\mathrm{A}
\end{aligned}
$
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