Advanced Linear Algebra, Lecture 3.5: The determinant and trace of a matrix

Advanced Linear Algebra, Lecture 3.5: The determinant and trace of a matrix

We begin this lecture by using n-linearity to give explicit formulas for the determinant of a 2x2 and 3x3 matrix, and it is easy to see how this generalizes to an nxn matrix. Specifically, the determinant is the sum of the determents of all n! permutation matrices, but with the original coefficients a_{ij} in the non-zero entries. This gives an easy way to see why a matrix and its transpose have the same determinant. We also see the definition of the trace of a matrix -- the sum of the diagonal entries. Though this can be defined for a general linear map, it involves eigenvalues, which we have not yet seen. We prove several basic properties of the trace: it is linear, it is commutative (i.e., tr(AB)=tr(BA)), and that similar matrices have the same trace and determinant.

[kyleyan]

amazing！It just helped me understood a question which I‘v been thinking about for a long time.

[fsaldan]

At 21' the sign of the permutation is missing.

[coldsunny]

Hey Professor Macauley, do you mind if you can help me with a combinatorial proof question? its really troubling to understand