Solving 2x2 systems using inverse matrix (December 12, 2013)
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This document provides learning objectives and examples for representing systems of linear equations in matrix form. It defines matrix form, shows how to write 2x2 systems of equations in matrix form as AX = B, and explains how to find the inverse of a 2x2 matrix A-1 to solve the system. An example problem at the end represents a real-world scenario about revenue at two restaurant branches in matrix form to solve for the price of lunch and dinner sets.
Solving 2x2 systems using inverse matrix (December 12, 2013)
- 2. Learning Objectives To define matrix form To name the elements of matrix form To represent a 2x2 system of linear equations in matrix form To calculate the determinant of A, det(A) To calculate the inverse of A, i.e. A 1 , if det(A)≠0 To solve the 2x2 system of linear equations by using inverse matrix method
- 3. Consider this.. Taldy Bear Restaurant has 2 branches. Branch M is located in a mall and Branch G is located nearby a gym. Both branches sell lunch and dinner sets to public everyday. Last Sunday, Branch M made a revenue of 700,000 Tenge by selling 600 lunch sets and 1400 dinner sets. Branch G attained a revenue of 450,000 Tenge by selling 700 lunch sets and 350 dinner sets. 1. Fill in the blanks in the table below. Taldy Bear Restaurant Lunch Dinner Revenue (in Tenge) 600 1400 1,320,000 Branch M Branch G 700 350 577,500 2. Let x represents the price of one lunch set and y represents the price of one dinner set. Set up 2x2 system of equations that represents the above information x y x y 600 1400 1,320,000 700 350 577,500
- 4. Matrix Form Any 2x2 system of linear equations can be written in matrix form. Example: x y x y 2 7 3 4 17 a 2x2 system of linear equations x y 1 2 7 3 4 17 Matrix form: Coefficient matrix unknowns constants AX B
- 5. Give it a try Express each of the following systems of equations in matrix form x y x y 3 5 13 2 7 81 x y x y 3 4 6 3 4 18 x y 3 5 13 2 7 81 x y 3 4 6 3 4 18
- 6. Inverse of 2x2 Matrix Suppose A is a 2x2 matrix. The inverse of a 2x2 matrix A, denoted by , is a 2x2 matrix such that Hence, If , then 1 A 1 1 AA A A I 1 1 d b A ad bc c a a b A c d Determinant of matrix A Note: If the determinant of matrix is zero, then doesn’t exist. 1 A 1 1 1 AX B A AX A B X A B Matrix Form Solution of the system in term of A-1
- 7. Practice makes perfect 1. Which one of the following matrices does not have an inverse? 1 2 1 2 1 2 , , 3 5 3 4 3 6 2. Express the system of equations in matrix form. Determinant (1)(6)-(3)(2)=0 x y x y 2 7 3 4 17 Answer: (3,2) Find the inverse of the coefficient matrix. Solve the system for x and y. 3. Express the system of equations in matrix form. x y x y 5 13 3 2 5 Answer: (3,-2) Find the inverse of the coefficient matrix. Solve the system for x and y.
- 8. Back to Taldy.. Last Sunday, Branch M made a revenue of 700,000 Tenge by selling 600 lunch sets and 1400 dinner sets. Branch G attained a revenue of 450,000 Tenge by selling 700 lunch sets and 350 dinner sets. Taldy Bear Restaurant Lunch Dinner Revenue (in Tenge) 600 1400 1,320,000 Branch M Branch G 700 350 577,500 Let x represents the price of one lunch set and y represents the price of one dinner set. Find the values of x and y. x y x y 600 1400 1,320,000 700 350 577,500 x y 600 1400 1,320,000 700 350 577,500 Answer: x=450 Tenge y=750 Tenge