Maybe I shouldn't have started this blog now, not with everything that's been going on.

One thing I think I forgot to mention before about one of my previous lectures in Advanced Mathematics. After the lecture on matrix multiplication, there was a rationale for the “motivation” of that operation.

For this one, the handout recommended imagining three coordinate systems on the same (obviously two-dimensional) plane.

For simplicity’s sake, I assigned them to be A1 and A2, which coincide with the regular x and y-axes. B1 and B2 are tilted 30 degrees to the first two. C1 and C2 are tilted 30 degrees further.

Thus, having the equation of a line expressed along the A1-A2 axes would have different coefficients on the B axes, and on the C axes.

I just introduced the concept though, and have yet to provide an example, but I plan to, and that is a start. After all, it won’t be difficult to lead eight people in the dark, at least two or three at a time, and still be assured that they all get out of the tunnel.

Third trimester, third week of classes, first meeting, Advanced Mathematics: I first discussed the obtaining the determinants of matrices.

What surprised me was that they never used it for solving systems of equations in two or more variables, although they did use substitution, elimination and even graphical method.

In fact, some of the students were bemoaning that determinants was an easier method to use and could have helped them greatly during their engineering mechanics classes with David in the past two terms.

Next topic was the co-factor method of computing for determinants, which is for matrices whose order is four and above, by adding the products of the elements of one row with its minor. The minor of a matrix element is the sub-matrix involving all the elements not in its row and column.

I also showed them the properties of determinants of matrices, which includes the multiplication by a scalar (only to one column, unlike the scalar multiplication of a matrix). I also showed them that the determinant is equal even for the transpose of a matrix.

We illustrated by an example getting the negative of the determinant of the matrix when there is a switching of values between any two columns or rows. I proved that the determinant is zero if any row or column is completely proportional (or a multiple of or equal to) another row or column.

If all the elements of one row or column is replaced by the sum of that element and the product of the element of another row and a constant, the determinant will still be the same. Lastly, the determinant of the multiplication of two matrices is equal to the multiplication of their determinants, which is also the same as the determinant of the non-commutative multiplication of the matrices.

There are two not so flattering asides about this lecture I would like to mention, but that will have to wait until next Monday. For now, we’re finished for this week.