Determinants and What They Measure
The determinant is the factor by which a map scales area or volume, with a sign for orientation. It vanishes exactly when the map collapses space, which is exactly when no inverse exists.
What a learner can do afterwards
- Compute a determinant by cofactor expansion and by row reduction
- Interpret a determinant as an area or volume scale factor
- Connect a zero determinant to a system with no unique solution
1 · Read
For a 2 by 2 matrix with rows (a, b) and (c, d), the determinant is a times d minus b times c. A triangular matrix has only zeros below the diagonal, or only zeros above it, and its determinant is the product of the diagonal entries. In a 3 by 3 matrix one zero below the diagonal is not enough: all three entries below it must be zero, or all three above it.
Cofactor expansion handles bigger matrices. Go along row 1 of a 3 by 3 matrix. Multiply each entry by the determinant of the 2 by 2 block left when you cover that entry's row and column, and attach the signs plus, minus, plus. Row reduction also works: adding a multiple of one row to another keeps the determinant, swapping two rows flips its sign, and a triangular result gives the diagonal product.
The determinant is the factor by which a map scales area or volume. Its size tells you how much regions grow or shrink. Its sign tells you about orientation: positive keeps it, negative flips it. A determinant of 2 doubles every area and keeps orientation.
Stack the vectors (2, 1) and (1, 3) as the columns of a matrix, so its rows are (2, 1) and (1, 3). The determinant is 2 times 3 minus 1 times 1, which is 5. The parallelogram they span has area 5, the absolute value of the determinant. For a 3 by 3 matrix, the absolute determinant gives the volume of the slanted box its columns span.
Read the determinant before you solve a square system. If it is nonzero, the map can be undone, so there is exactly one solution. If it is zero, the map collapses space: the rows depend on each other, there is no inverse, and the system has either no solution or infinitely many, never exactly one.
Compute with a times d minus b times c, cofactor expansion or a triangular diagonal product, read the size as an area scale, and treat zero as collapse.
The rewritten readings were written for this dataset. Video by Khan Academy. Practice questions are original.
2 · Watch
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Where it sits
14 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.