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

For the first meeting of my Advanced Mathematics students on the fifth week of classes, we continued with the solutions for systems of linear equations using matrices.

Before we had taken up determinants. Now I taught them the Gauss Elimination method, which involved using one of the properties of determinants that we discussed before their first exam.

This property was the one that stated that if all the values of one row or column of a matrix is added to the elements of another row or column multiplied by a constant, and then the determinant of the new matrix would be the same as the original one.

They could use this property to be able to make the values of certain elements equal to zero, and come up with a triangular matrix.

When the elements are returned to being coefficients of equations in n variables (n also counting as the order of the augmented square matrix) they will notice that for one equation, two of the variables are equal to zero and thus they are left with one variable and a constant.

This they can substitute in the second equation, which now only has two variables. The two values they replace in the third equation with three variables.

There was a variation of this elimination method, named after Gauss and another guy named Jordan.

For this one they had to make a diagonal matrix, which required more property manipulation of elements than the first method to turn their values to zero.

But there was no more substitution afterwards, and only transposition of constants.

After that I gave them an exercise that was only supposed to last for 20 minutes, but one pair (out of the three I assigned) could not finish the method I tasked to them, even though they could check their answers with the other teams.

That meant that our next topic, about the inversion of matrices, had to be postponed.

I only got to discuss the property of the identity matrix, and a demonstration with a two by two matrix. I also gave the definition of the inverse, and again showed an example of how to solve for the elements of the inverse of a two by two matrix, by using four variables.

Despite this, they only ended up with two sets of two equations with two unknowns each, making it easy to solve.

Of course they asked that we multiply the original matrix with its inverse to see if it would really result in an identity matrix.

Then I extended the problem to a three by three matrix, and told them they could get the matrix by using nine variables, which they obviously were averse to.

I’ll continue this tale next week. For now, class dismissed.