Matrices in Real Computation

A shop mixes cashews ($8/kg)andpeanuts($3/kg) to make $10$ kg of mix costing $58 in total. How many kg of each are used?

• 1Let $x$ = kg of cashews, $y$ = kg of peanuts. Set up equations: $x+y=10$, and $8x+3y=58$
• 2Write as a matrix equation: A = $\left(\begin{array}{cc}1& 1\\ 8& 3\end{array}\right)$, B = $\left(\begin{array}{c}10\\ 58\end{array}\right)$
• 3det(A) = ($1\times 3$) − ($1\times 8$) = $3-8=-5$. So $A^{-1}$ = ($1÷-5$) × $\left(\begin{array}{cc}3& -1\\ -8& 1\end{array}\right)=\left(\begin{array}{cc}-0.6& 0.2\\ 1.6& -0.2\end{array}\right)$
• 4Multiply $A^{-1}$ by B: $x$ = ($-0.6\times 10$) + ($0.2\times 58$) = $-6+11.6=5.6$. $y$ = ($1.6\times 10$) + ($-0.2\times 58$) = $16-11.6=4.4$

Answer: $5.6$ kg cashews, $4.4$ kg peanuts. Check: $5.6+4.4=10$ ✓, and $8$($5.6$) + $3$($4.4$) = $44.8+13.2=58$ ✓