Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Quick answer: A system of linear equations can be written as AX = B, where A holds the coefficients. When det A ≠ 0, the unique solution is X = A⁻¹B. A 2 × 2 matrix also transforms the plane, and the absolute value of its determinant is the factor by which areas change.
Standards: A-REI.C.8, A-REI.C.9, N-VM.C.11, N-VM.C.12 · Level: Grade 12 Precalculus
How do you write a system as a matrix equation?
A pair of linear equations can be stacked into one statement, AX = B. The coefficients form the matrix A, the unknowns form a column X, and the constants form a column B. For 5x − 2y = 22 and x + 3y = 1:
A has rows (5, −2) and (1, 3). X is the column with entries x and y. B is the column with entries 22 and 1. The statement AX = B then says the same thing as both equations together.
How do you solve it with an inverse matrix?
- Compute det A = ad − bc. If it is 0, there is no unique solution.
- Build A⁻¹ by exchanging the diagonal, flipping the off-diagonal signs and dividing by det A.
- Multiply on the left: X = A⁻¹B.
- Substitute the answer into both equations.
Here det A = 5(3) − (−2)(1) = 17, so A⁻¹ = 117 times rows (3, 2) and (−1, 5). Multiplying by the column (22, 1) gives 117(3·22 + 2·1, −22 + 5) = 117(68, −17) = (4, −1). So x = 4 and y = −1.
Computing BA⁻¹. Matrix products depend on order, and the inverse must multiply B from the left. For three or more unknowns, use technology to find the inverse.
How do 2 × 2 matrices transform the plane?
Multiplying a matrix by a point written as a column moves the point. Each column of the matrix tells where the points (1, 0) and (0, 1) land.
| Transformation | Matrix rows | Effect on (x, y) |
|---|---|---|
| Rotation 90° counterclockwise | (0, −1) and (1, 0) | (−y, x) |
| Reflection in the x-axis | (1, 0) and (0, −1) | (x, −y) |
| Reflection in the y-axis | (−1, 0) and (0, 1) | (−x, y) |
| Scale by k | (k, 0) and (0, k) | (kx, ky) |
| Horizontal shear | (1, k) and (0, 1) | (x + ky, y) |
What does the determinant tell you about area?
The absolute value of the determinant is the factor by which a transformation scales area. If det = 2, every shape doubles in area. A negative determinant reverses orientation, and det = 0 flattens the plane onto a line. For the matrix with rows (2, 1) and (1, 3), det = 5, so the unit square becomes a parallelogram of area 5.
To build a transformation matrix, track where (1, 0) and (0, 1) go and write those images as the two columns.
- Write a system as AX = B: coefficient matrix, unknown column, constant column.
- If det A ≠ 0, then X = A⁻¹B; multiply on the left. If det A = 0, there is no unique solution.
- A 2 × 2 matrix times a point written as a column gives the transformed point.
- The columns of a transformation matrix are the images of (1, 0) and (0, 1).
- Areas scale by |det|; a negative determinant also flips orientation.
Solving Systems and Transforming the Plane: frequently asked questions
Why must the inverse multiply from the left?
Because AX = B multiplies X by A on the left. To cancel A, multiply both sides by A⁻¹ on the left, giving X = A⁻¹B. Multiplying on the right is not a valid step and usually produces a product that is not even defined.
What if det A is zero?
Then A has no inverse. The lines are either parallel, so there is no solution, or the same line, so there are infinitely many. Row-reducing or substituting tells which case you have.
How do I solve a 3 × 3 system with matrices?
Write AX = B with a 3 × 3 matrix A, then use a calculator or software to compute A⁻¹ and multiply by B. The idea is identical to the 2 × 2 case; only the arithmetic grows.
How do I find the matrix of a transformation?
Apply the transformation to (1, 0) and (0, 1). Write the image of (1, 0) as the first column and the image of (0, 1) as the second column. Every other point follows by linearity.
What does a negative determinant mean for a transformation?
The area still scales by the absolute value of the determinant, but the shape is flipped like a mirror image, as with a reflection. A reflection has determinant −1.
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