DeterminantsIntroduction
At the school mela, Aman runs a guessing game. He thinks of two numbers and hands Riya two clues: "the first number plus twice the second is 4" and "twice the first minus the second is 3". Riya writes them as equations, and , and draws each one as a line. Every point on a line fits that clue, so the point where the two lines cross fits both: Aman's numbers are 2 and 1.
Clues can be badly chosen, though. Sometimes two clues can never both be true, and sometimes the second clue says nothing new. Is there a quick test that tells you which, before you draw anything?
You are Aman now. The blue line is clue (1) and stays put; drag the two green handles to change the green line, clue (2). Under the plane, keep an eye on the number — it is built from the four coefficients of x and y. Complete both missions.
Both times the pair failed to have exactly one solution, the number dropped to 0 — and whenever the lines crossed at a single point, it was not 0. In general, the pair , has a unique solution exactly when .
Notice that the number uses only the coefficients, arranged as the square
Discovered Twice
Determinants were discovered twice, at almost the same time, on opposite sides of the world. In 1683 the Japanese mathematician
The word came later.
Drag these milestones into the order in which they happened, earliest at the top.
In this chapter you will evaluate determinants of order up to three, use them to find areas of triangles, meet minors and cofactors, build the adjoint and the inverse of a square matrix, and finally solve systems of linear equations — and decide when a system has no solution at all.