3.4 Determinants
Readings
At this time please read Chapter 3 Section 4 in Understanding Linear Algebra by David Austin.
The determinant of an nxn matrix can be thought of as measuring the area (in the 2×2 case) or volume (in the 3×3 case) of the parallelogram (in the 2×2 case) or box/parallelepiped (in the 3×3 case) formed by the column vectors, together with a sign measuring the orientation of these vectors. We can also think of it as the factor by which area / volume is multiplied when we apply a linear transformation.
Key Takeaways
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- det(AB) = det(A)det(B)
- If A is an nxn matrix, then [latex]det(kA) =k^n det(A)[/latex]
- A is invertible if and only if [latex]det(A) \neq 0[/latex],
Computing determinants examples:
Row reduction operations and their effect on the determinant:
