To solve the problem using matrix methods, we first need to set up equations based on the provided information.

1. Define the Variables

Let:

• $a$ = number of Cointronic cocktails consumed by Anton
• $b$ = number of Dry Martini cocktails consumed by Anton
• $c$ = number of Dry Martini cocktails consumed by Beat
• $d$ = number of Margarita cocktails consumed by Beat
• $f$ = number of Sex on the Beach cocktails consumed by Claus
• $g$ = number of Margarita cocktails consumed by Claus
• $h$ = number of Tequila Sunrise cocktails consumed by Claus

2. Set Up the Alcohol Content Equations

Alcohol content of each cocktail:

• Cointronic: $4 \, \text{cl gin} \times 0.375 + 1 \, \text{cl orange liqueur} \times 0.40 + 20 \, \text{cl tonic water} \times 0 = 1.5 + 0.4 = 1.9 \, \text{cl}$
• Dry Martini: $5 \, \text{cl gin} \times 0.375 + 8 \, \text{cl Martini Extra Dry} \times 0.15 + 1 \times 0 = 1.875 + 1.2 = 3.075 \, \text{cl}$
• Sex on the Beach: $4 \, \text{cl vodka} \times 0.40 + 2 \, \text{cl peach liqueur} \times 0.21 + 4 \, \text{cl orange juice} \times 0 + 4 \, \text{cl cranberry juice} \times 0 = 1.6 + 0.42 = 2.02 \, \text{cl}$
• Margarita: $1 \, \text{cl lime juice} \times 0 + 2 \, \text{cl orange liqueur} \times 0.40 + 3 \, \text{cl tequila} \times 0.38 = 0 + 0.8 + 1.14 = 1.94 \, \text{cl}$
• Tequila Sunrise: $6 \, \text{cl tequila} \times 0.38 + 2 \times 0 + 10 \, \text{cl orange juice} \times 0 + 0 = 2.28 + 0 = 2.28 \, \text{cl}$

3. Set Up the Equations

From the problem, we have:

• Anton's total alcohol: $1.9a + 3.075b = 8.775$
• Beat's total alcohol: $3.075c + 1.94d = 8.895$
• Claus's total alcohol: $2.02f + 1.94g + 2.28h = 10.54$

4. Write the Matrix Form

We can write the system of equations in matrix form $Ax = b$: