A matrix stores a transformation of the plane. Multiplying a vector by it applies that transformation.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Multiplying
Take each row of the matrix against the vector, multiplying matching entries and adding.
Order matters
Matrix multiplication is not commutative. Rotating then stretching is not the same as stretching then rotating.
The determinant
For [[a, b], [c, d]] it is ad − bc, and it is the factor by which areas are scaled.
A determinant of zero
Zero means the transformation flattens the plane onto a line, destroying information. No inverse exists.
The inverse
An inverse matrix undoes the transformation. It exists exactly when the determinant is not zero.
A grid of numbers that moves the plane
Multiplying a vector by a 2×2 matrix rotates, scales, shears or reflects it. The matrix is a transformation written in numbers, and its columns are the images of the two unit vectors.
Multiplication composes transformations
Applying one matrix and then another is the same as applying their product. Since the order matters, matrix multiplication is not commutative — which is a direct consequence of the geometry.
The determinant measures area scaling
It gives the factor by which areas are multiplied, with a negative sign if orientation is reversed. A determinant of zero collapses the plane onto a line, which is why such a matrix has no inverse.
The inverse undoes the motion
A matrix times its inverse gives the identity, leaving every vector where it was. The inverse exists precisely when the determinant is nonzero, and it is how a transformation is reversed.
Step 2: Try It Yourself
Tap and try it out.
- Determinant of M4
A non-zero determinant means the matrix is invertible. Its size is the factor by which areas are scaled — here 4.
Step 3: Watch an Example
One step at a time.
Watch Marcus Read a Determinant
Marcus has the matrix [[3, 1], [2, 4]] and a shape of area 5.
- Step 1
The determinant is 3 × 4 − 1 × 2 = 12 − 2.
Step 4: Your Turn
Practice makes it stick.
The Determinant
Problem 1 of 2
[[3, 1], [2, 4]]. What is the determinant?
The Area
Problem 2 of 2
A shape of area 5 under a transformation with determinant 10. What is the new area?
Transform the Plane
1 of 8
[[5, 2], [3, 4]]. Determinant?
2 of 8
[[1, 0], [0, 1]]. Determinant?
3 of 8
[[6, 3], [4, 2]]. Determinant?
4 of 8
Does that matrix have an inverse? 1 yes, 0 no.
5 of 8
Determinant 4 applied to a shape of area 7. New area?
6 of 8
Row (1, 2) against the vector (5, 7). What is the result?
7 of 8
Match each determinant value with what it means.
Tap a card on the left to start.
8 of 8
[[7, 3], [2, 1]]. Determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
[[4, 1], [3, 2]]. What is the determinant?
Question 2 of 2
What does a determinant of zero mean?
What You Learned
- A matrix stores a transformation, applied by multiplication.
- The determinant is the area scaling factor.
- A zero determinant collapses the plane and leaves no inverse.