Determinant
A single number computed from a square matrix. For [[a,b],[c,d]] it is ad - bc. It tells you how much the matrix scales areas.
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Analogy
A stretch factor: how many times bigger a shape becomes after the matrix acts on it.
Example
det = 3(4) - 1(2).= 12 - 2.= 10.A unit square becomes area 10.
In practice
A zero determinant means the matrix squashes space flat and cannot be undone.
Practice
Flashcards
- What is the determinant of [[a,b],[c,d]]?ad - bc.
- What is the determinant of [[3,1],[2,4]]?10.
- What does a zero determinant mean?The matrix has no inverse.
Questions people ask
What does a negative determinant mean?
The matrix flips space over, like looking in a mirror, as well as scaling it. The size of the number is still the area scale factor. A determinant of -10 means areas grow tenfold and the orientation is reversed. Swapping two rows of a matrix has this same effect on its determinant.
What does a determinant of zero tell me?
The matrix squashes the plane onto a line or a point, so some area is lost and cannot be recovered. That means the matrix has no inverse, and a system with that matrix does not have exactly one solution. Instead, it has none or infinitely many.