Question
If $A=\left[\begin{array}{cc}2 & -3 \\ 3 & -2 \\ -1 & 4\end{array}\right], B=\left[\begin{array}{ccc}-3 & 4 & 1 \\ 2 & -1 & -3\end{array}\right]$, verify
$\left(3 A-5 B^{\top}\right)^{\top}=3 A^{\top}-5 B$
$\left(3 A-5 B^{\top}\right)^{\top}=3 A^{\top}-5 B$
$=\left[\begin{array}{cc}6 & -9 \\ 9 & -6 \\ -3 & 12\end{array}\right]-\left[\begin{array}{cc}-15 & 10 \\ 20 & -5 \\ 5 & -15\end{array}\right]$
$\therefore \quad 3 A-5 B^{\top}=\left[\begin{array}{cc}21 & -19 \\ -11 & -1 \\ -8 & 27\end{array}\right]$
$\therefore \quad\left(3 A-5 B^{\mathrm{T}}\right)^{\mathrm{T}}=\left[\begin{array}{ccc}21 & -11 & -8 \\ -19 & -1 & 27\end{array}\right]$
$\ldots$..i)
$3 A^T-5 B=3\left[\begin{array}{ccc}2 & 3 & -1 \\ -3 & -2 & 4\end{array}\right]-5\left[\begin{array}{ccc}-3 & 4 & 1 \\ 2 & -1 & -3\end{array}\right]$
$=\left[\begin{array}{ccc}6 & 9 & -3 \\ -9 & -6 & 12\end{array}\right]-\left[\begin{array}{ccc}-15 & 20 & 5 \\ 10 & -5 & -15\end{array}\right]$
$=\left[\begin{array}{ccc}21 & -11 & -8 \\ -19 & -1 & 27\end{array}\right]$
...(ii)
From (i) and (ii), we get
$\left(3 A-5 B^T\right)^T=3 A^T-5 B$
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f(x) = x3 – 2x + 1, if x ≤ 2
= 3x – 2, if x > 2, at x = 2.