Review and Practice

Let A = $\left(\begin{array}{cc}2& 3\\ 1& 4\end{array}\right)$. Find det(A), find $A^{-1}$, then use it to solve $2x+3y=13$ and $x+4y=14$.

• 1det(A) = ($2\times 4$) − ($3\times 1$) = $8-3=5$
• 2$A^{-1}$ = ($1÷5$) × $\left(\begin{array}{cc}4& -3\\ -1& 2\end{array}\right)=\left(\begin{array}{cc}0.8& -0.6\\ -0.2& 0.4\end{array}\right)$
• 3Multiply $A^{-1}$ by B = $\left(\begin{array}{c}13\\ 14\end{array}\right)$: $x$ = ($0.8\times 13$) + ($-0.6\times 14$) = $10.4-8.4=2$
• 4$y$ = ($-0.2\times 13$) + ($0.4\times 14$) = $-2.6+5.6=3$

Answer: det(A) = $5$, $x=2$, $y=3$. Check: $2$($2$) + $3$($3$) = $4+9=13$ ✓, and $2+4$($3$) = $2+12=14$ ✓

M = $\left(\begin{array}{cc}0& 2\\ -2& 0\end{array}\right)$. Apply M to the point ($1$, $3$), and find the area scale factor of M.

• 1Apply M: top entry = ($0\times 1$) + ($2\times 3$) = $6$. Bottom entry = ($-2\times 1$) + ($0\times 3$) = $-2$. New point: ($6$, $-2$)
• 2det(M) = ($0\times 0$) − ($2\times -2$) = $4$
• 3Area scale factor = |$4$| = $4$

Answer: New point = ($6$, $-2$), area scale factor = $4$