The price of a hot dog is x and the price of a drink is y. We will first write the verbal expressions in the exercise as algebraic expressions.
Verbal Expression
Algebraic Expression
Case I
Two hot dogs and one drink cost $ 4.50.
2 x+ 1 y= 4.50
Case II
Three hot dogs and two drink cost $ 7.25.
3 x+ 2 y= 7.25
To evaluate the values of x and y now we want to find the multiplier matrix of the following matrix.
4.50
7.25
To do so we will write a system of equations by using the equations that we wrote. Then we will rewrite the system as a matrix equation.
2 x+ 1 y= 4.50 3 x+ 2 y= 7.25
⇓
2 & 1
3 & 2
x
y
=
4.50
7.25
Recall that the left-hand side of the above matrix equation is the product of the coefficient matrix and the variable matrix, while the right-hand side is the constant matrix. We will isolate the variable matrix by multiplying both sides by the inverse of the coefficient matrix. When multiplying a matrix by its inverse, we obtain the identity matrix.
A^(- 1)=1/ad-bc [ cc d & - b - c & a ] where ad-bc ≠0
The expression ad-bc is known as the determinant of a 2* 2 matrix. Because it is in the denominator of a fraction, if the determinant is zero the matrix cannot have an inverse. Consider our coefficient matrix.
[
cc
2 & 1 3 & 2
]
Let's calculate its determinant.
Since the determinant is not zero, the matrix has an inverse. We can now apply the formula for the inverse. Note that we usually refer to the determinant using the notation ad-bc=det(A).
