Question3) Given $A=\left[\begin{array}{cc}3 & -5 \\ -1 & 7\end{array}\right]$ and $B=\left[\begin{array}{ll}1 & 4 \\ 8 & 3\end{array}\right]$ find the following a) $A+B$ b) $A B$ c) $B^{\top} A^{\top}$ d) $(A B)^{\top}$ e) $2(A-B)$

Studdy Solution
Compute the scalar multiplication $2(A - B)$.
$A - B = \left[\begin{array}{cc} 3 & -5 \\ -1 & 7 \end{array}\right] - \left[\begin{array}{cc} 1 & 4 \\ 8 & 3 \end{array}\right] = \left[\begin{array}{cc} 3-1 & -5-4 \\ -1-8 & 7-3 \end{array}\right]$
$A - B = \left[\begin{array}{cc} 2 & -9 \\ -9 & 4 \end{array}\right]$
Now multiply by 2:
$2(A - B) = 2 \left[\begin{array}{cc} 2 & -9 \\ -9 & 4 \end{array}\right] = \left[\begin{array}{cc} 4 & -18 \\ -18 & 8 \end{array}\right]$
Solutions: (a) $$A + B = \left[\begin{array}{cc} 4 & -1 \\ 7 & 10 \end{array}\right]$$ (b) $$A B = \left[\begin{array}{cc} -37 & -3 \\ 55 & 17 \end{array}\right]$$ (c) $$B^{\top} A^{\top} = \left[\begin{array}{cc} -37 & 55 \\ -3 & 17 \end{array}\right]$$ (d) $$(A B)^{\top} = \left[\begin{array}{cc} -37 & 55 \\ -3 & 17 \end{array}\right]$$ (e) $$2(A - B) = \left[\begin{array}{cc} 4 & -18 \\ -18 & 8 \end{array}\right]$$