For rows [a, b] and [c, d], the determinant is ad − bc. The formula is easy; what it measures is the point.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
It is an area factor
The determinant is the factor by which the transformation multiplies every area. A determinant of 3 triples every region.
Where to see it
The unit square has area 1. Its image is a parallelogram, and that parallelogram has area equal to the determinant.
The sign
A negative determinant means the plane was flipped over. The size still gives the area factor.
Zero
A determinant of zero means the parallelogram collapsed to a segment. Everything has been squashed onto a line.
The connection to inverses
Zero determinant means dependent columns, no inverse, and either no solution or infinitely many. All of those are one fact.
How much space was stretched
The determinant gives the factor by which the transformation multiplies area in two dimensions, or volume in three. A determinant of 3 triples every area in the plane.
The sign records orientation
A negative determinant means the transformation flips orientation, as a reflection does. So the determinant carries both a scaling factor and a handedness.
Zero means collapse
A determinant of zero means everything is squashed onto a lower-dimensional set — a line or a point. Area becomes zero, information is lost, and no inverse can exist.
Computing it
The 2×2 case is ad − bc. Larger matrices use cofactor expansion or, far more efficiently, row reduction. For anything beyond 3×3, expansion by cofactors is impractically slow.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(3, 1)
- j-hat lands on(1, 2)
- Determinant5
The shaded parallelogram is the image of the unit square, and its area is 5. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Bassam Read an Area
Bassam applies the matrix with rows [3, 1] and [1, 2] to a shape of area 4.
- Step 1
He computes the determinant: 6 − 1 = 5.
Step 4: Your Turn
Practice makes it stick.
The Stretch
Problem 1 of 2
A determinant of 6 applied to a shape of area 3. What is the new area?
The Flip
Problem 2 of 2
A determinant of −4 applied to a shape of area 5. What is the new area?
Measure the Stretch
1 of 8
Rows [4, 1] and [2, 3]. What is the determinant?
2 of 8
Rows [2, 6] and [1, 3]. What is the determinant?
3 of 8
Rows [5, 0] and [0, 5]. What is the determinant?
4 of 8
A determinant of 0. What is the area of the image of the unit square?
5 of 8
A determinant of 7 on a shape of area 2. What is the new area?
6 of 8
Rows [1, 2] and [3, 4]. What is the determinant?
7 of 8
Sort each determinant by what it says about the transformation.
Tap something to move it.
- Empty
- Empty
8 of 8
The identity matrix. What is its determinant?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Rows [6, 2] and [4, 3]. What is the determinant?
Question 2 of 2
What does the determinant measure?
What You Learned
- The determinant ad − bc is the factor by which a transformation scales every area.
- Its sign records whether the plane was flipped over.
- A determinant of zero means everything collapsed onto a line, so no inverse can exist.