Monday, August 30, 2010

Even the Mona Lisa smiles behind the blackboard.

Beauty that is rare like a painting of the Mona Lisa can be found behind the blackboard, in front of a mathematics equation.
I had the “Oh my, what beauty!” experience while solving a well worn mathematics problem a new way. The problem: simultaneous equations in two or more variables.
I’m sure you do remember the age-old technique of solving simultaneous equations in two unknowns.
First, chose one of the variables as the one to exterminate. Multiply both equations, numbered of course, by suitable coefficients so that the variable that will be exterminated becomes the same in coefficients. You then subtract one from the other, giving you an equation in a unique unknown. Finally, you solve the unique unknown. Your answer for this single unknown can then be used to find the second unknown variable using one of the simultaneous equations as template.
Hmm! Very long process that!
Matrices is easier. That is where the beauty lies. Take the value of the second order determinants for the two unknowns; call this the raw determinant. For each unknown, replace the column for that unknown by the column for solution to the equations. Take the value of the determinants for each unknown whose column has been substituted. Then to get the value of that unknown, divide the value of the determinants for the substituted column by the raw determinant.
Easier. Faster. More energy efficient.
Don’t believe the above line until you get equations of more than two unknowns. I saw the Mona Lisa smiling behind the blackboard and it was a matrix blackboard with blue markers.
I’ll be taking calculus with my students tomorrow.

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