Any linear system can be written AX = B: A holds the coefficients, X the unknowns and B the constants.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Solving it
Multiplying both sides by A⁻¹ gives X = A⁻¹B. It is the matrix version of dividing both sides.
Which side to multiply on
Multiply on the left of both sides. Because matrix multiplication is not commutative, A⁻¹B and BA⁻¹ are different.
Cramer’s Rule
Each unknown equals a determinant ratio: replace the matching column of A with B, take that determinant, and divide by the determinant of A.
When the determinant is zero
A zero determinant means the system has no unique solution: the lines are either parallel or identical.
Why bother
For two equations, substitution is faster. For twenty, the matrix method is what a computer actually runs.
A whole system as one equation
A system of linear equations becomes AX = B, where A holds the coefficients, X the unknowns and B the constants. The entire system collapses into a single matrix equation.
Solving by inverse
If A is invertible, X = A⁻¹B. One matrix inversion solves the system for any right-hand side, which is why this form is preferred when the same coefficients recur with different constants.
A zero determinant means no unique solution
If det A is zero the system has either no solution or infinitely many. The determinant diagnoses the case before you attempt to solve, which saves discovering it halfway through elimination.
Why this matters at scale
Real problems involve systems with thousands of equations — structural analysis, circuit simulation, economic models. Matrix methods are how those are solved, and hand calculation on 2×2 systems is only the introduction.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of M5
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 5.
Step 3: Watch an Example
One step at a time.
Watch Diego Solve a System
Diego has 2x + y = 8 and x + 3y = 9.
- Step 1
The coefficient matrix is [[2, 1], [1, 3]], and its determinant is 6 − 1 = 5.
Step 4: Your Turn
Practice makes it stick.
The Determinant
Problem 1 of 2
A system has coefficient matrix [[3, 2], [1, 4]]. What is its determinant?
The Tickets
Problem 2 of 2
Adult tickets cost $8 and child tickets $5. 10 tickets cost $68. How many adult tickets were sold?
Systems as Matrices
1 of 8
Coefficient matrix [[1, 2], [3, 4]]. Determinant?
2 of 8
Coefficient matrix [[2, 4], [1, 2]]. Determinant?
3 of 8
With that zero determinant, is there a unique solution? 1 for yes, 0 for no.
4 of 8
In AX = B, what does X hold?
5 of 8
Determinant of A is 5, and the x-determinant is 20. What is x?
6 of 8
Determinant of A is 4, and the y-determinant is −8. What is y?
7 of 8
Put the matrix solving method in order.
- 1Compute the determinant of A.
- 2Confirm it is not zero.
- 3Multiply both sides on the left by A⁻¹.
- 4Write the system as AX = B.
8 of 8
x + y = 7 and x − y = 1. What is x?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Determinant of A is 3, and the x-determinant is 12. What is x?
Question 2 of 2
A coefficient matrix has determinant zero. What does that say about the system?
What You Learned
- 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.