Lesson 6.3

 Three friends, Duane, Marsha, and Parker, decide to take their younger siblings to the movies. Before the movie, they buy some snacks at the concession stand. ◦ Duane buys two candy bars, a small drink, and two boxes of chocolate-covered peanuts for a total of $ ◦ Marsha spends $9.00 on a candy bar, two small drinks, and one box of chocolate-covered peanuts. ◦ Parker spends $12.35 on two small drinks and three boxes of chocolate-covered peanuts, but doesn’t buy any candy bars.  If all the prices include tax, what is the price of each item?

 Let c represent the price of a candy bar in dollars  let d represent the price of a small drink in dollars  let p represent the price of a box of chocolate-covered peanuts in dollars.  This system represents the three friends’ purchases:

 To use matrices to solve this system first, translate these equations into a matrix equation in the form [A][X]=[B]

 Solving this equation [A][X]=[B] is similar to solving the equation ax=b.  We will multiply both sides by the inverse of a or a -1 Where [I] in an identity matrix and [A] -1 in an inverse matrix.

 Let’s find an identity matrix for An identity matrix for matrix will be another 2 x 2 matrix so that

Multiplying the two matrices yields

 Because the two matrices are equal, their entries must be equal. This yields:  By using substitution and elimination you can find that a = 1, b = 0, c = 0, and d = 1.  Therefore

 The 2 x 2 identity matrix is Can you see why multiplying this matrix by any 2 x 2 matrix results in the same 2 x 2 matrix?

Identity Matrix [I]: An identity matrix [I] is a square matrix that does not change another square matrix when multiplied. If [A] is a given square matrix then [I] is an identity matrix if

Inverse Matrix [A] -1 : If [A] is a square matrix then [A] -1 is the inverse matrix of [A] if Where [I] is an identity matrix.

 In this investigation you will learn ways to find the inverse of a 2 x 2 matrix.  Use the definition of an inverse matrix to set up a matrix equation. Use these matrices and the 2 x 2 identity matrix for [I].

 Use matrix multiplication to find the product [A][A] -1. Set that product equal to matrix [I].

 Use the matrix equation from the previous step to write equations that you can solve to find values for a, b, c, and d.  Solve the systems to find the values in the inverse matrix. 2a+ c=1, 2b +d=0, 4a+ 3c=0, 4b+3d =1; a =1.5, b=0.5, c=2, d=1;

 Use your calculator to find [A ] -1. If this answer does not match your answer to the last step, check your work for mistakes.

 Find the products of [A][A] -1 and [A] -1 [A].  Do they both give you 1?  Is matrix multiplication always commutative?

 Not every square matrix has an inverse. Try to find the inverse of each of these matrices. Make a conjecture about what types of 2x2 square matrices do not have inverses. None of the matrices has an inverse. A 2 x 2 square matrix does not have an inverse when one row is a multiple of the other.

 Can a nonsquare matrix have an inverse? Why or why not? No. The product of a matrix and its inverse must be a square matrix because an identity matrix is always square and has the dimensions of the matrix and its inverse.

 Solve this system using an inverse matrix.  First, rewrite the second equation in standard form.

 The matrix equation for this system is  If this equations corresponds with find [A] -1.

The solution to the system is (2, 1). Substitute the values into the original equations to check the solution.

 Use an inverse matrix to solve the problem posed at the beginning of the lesson.  What is the cost of each snack item?