Cogito
Precalculus · Chapter 6 · Lesson 3
Solving Systems with Matrices
Write a whole system as one equation.
12 problems · about 22 minutes · TEKS P.3.H
What this lesson teaches
The student writes a linear system in matrix form and solves it using an inverse or Cramer’s Rule.
- A linear system is AX = B, solved by X = A⁻¹B.
- Cramer’s Rule finds each unknown as a ratio of determinants.
- A zero determinant means no unique solution exists.
Warm Up
Straightforward practice. Get the method working first.
5 problemsDeterminant of A is 3, and the x-determinant is 12. What is x?
Answer 4
Why 4.
A coefficient matrix has determinant zero. What does that say about the system?
Answer It has no unique solution.
Why No unique solution.
Coefficient matrix [[1, 2], [3, 4]]. Determinant?
Answer -2
Why 4 − 6.
Coefficient matrix [[2, 4], [1, 2]]. Determinant?
Answer 0
Why 4 − 4.
With that zero determinant, is there a unique solution? 1 for yes, 0 for no.
Answer 0
Why Cramer’s Rule would divide by zero.
Build It Up
The same ideas with more to keep track of.
3 problemsIn AX = B, what does X hold?
Answer The unknowns.
Why B holds the constants.
Determinant of A is 5, and the x-determinant is 20. What is x?
Answer 4
Why 20 ÷ 5.
Determinant of A is 4, and the y-determinant is −8. What is y?
Answer -2
Why −8 ÷ 4.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsPut the matrix solving method in order.
Answer 1. Write the system as AX = B. 2. Compute the determinant of A. 3. Confirm it is not zero. 4. Multiply both sides on the left by A⁻¹.
Why You cannot use an inverse before knowing one exists.
x + y = 7 and x − y = 1. What is x?
Answer 4
Why Add the two equations.
The Determinant: A system has coefficient matrix [[3, 2], [1, 4]]. What is its determinant?
Answer 10
Why 10.
The Tickets: Adult tickets cost $8 and child tickets $5. 10 tickets cost $68. How many adult tickets were sold?
Answer 6
Why 6 adult tickets.