Linear Algebra
Determinant
One number that says what a matrix does to area. The unit square becomes a parallelogram, and the determinant is its size, negative when the plane is turned over.
A î, the first columnA ĵ, the second columnpositive areanegative area, plane turned overafter B as well
Drag the tips of the two arrows
det A—
—
turned over−6flattened0kept+6
Matrix A
1.400.600.301.20
Presets
View
Then apply B as well
Show
Values
area of the square—
shape area, before → after—
det B—
det(BA)—
What to observe
- With base times height on, slide the second arrow along the first: the height never changes, so neither does the area. That is why adding a multiple of one row leaves det alone.
- Line the two arrows up and the parallelogram thins to nothing:det = 0, the plane is pressed onto a line, and A isnot invertible.
- Animate Reflect with the F shown. The F turns into its mirror image and det goes negative, and on the way it has to pass through zero.
- Apply B as well: area is scaled twice, so det(BA) = det B · det A. Two reflections multiply to +1 and the F comes back the right way round.
Shortcuts:space animate · r reset · f fullscreen