DERIVE

DETERMINANT DERIVATION FOR FEATHERMERCHANTS (10-second drumroll)

Given a system of two simultaneous (linear) algebraic equations ("two equations in two unknowns"):
a₁x + a₂y = c₁ [1]; b₁x + b₂y = c₂ [2]

We wish to reduce this to one equation in one "unknown", say, in x. To do so, start by multiplying all terms of equation [1] by the coefficient of y in equation [2], namely, by b₂. Then, multiply all terms of equation [2] by the coefficient of y in equation [1], namely, by a₂.
Result: a₁b₂x + a₂b₂y = b₂c₁[1']; a₂b₁x + a₂b₂y = b₁c₂[2']

Please notice that equations [1'], [2'] have the same second term, namely,a₂b₂y. Hence, subtracting equation [2'] from equation [1'] yields a single equation in "one unknown", namely, x, as proposed:
a₁b₂x — a₂b₁x = b₂c₁ — b₁c₂, or, collecting on x: (a₁b₂ — a₂b₁)x = b₂c₁ — b₁c₂[3]

We can now solve for x, alone, via dividing both sides of [3] by that parenthetical coefficient. But, TO OBTAIN A UNIQUE SOLUTION -- NOT "AN INFINITE NUMBER" -- we must ensure that we're not dividing by zero! Hence, we declare a₁b₂ ¹a₂b₁.
We now have: x = (b₂c₁ — b₁c₂)/(a₁b₂ — a₂b₁)[3']
Given the solution for one of the "unknowns", we can substitute it either [1] or [2] to solve for the other "unknown", finding (Assignment!): y = (a₂c₁ — a₁c₂)/(a₁b₂ — a₂b₁)[4]

Please notice that equations [3'], [4] have the same denominator term, namely, (a₁b₂ — a₂b₁).
And we noted above that THE VALUE OF THIS TERM DETERMINES WHETHER AN UNIQUE SOLUTION CAN BE OBTAINED. Hence, this kind of term has become known as "the determinant of the system of equations" or "the primary determinant". What is it? Looking at equations [1], [2], we see that it is a skew-form of the coefficients in [1], [2]. This patterns the commutator bracket for matrices, which do not commute under multiplication: [A, B] º AB — BA.
But this bracket does not allow for subscripts.
Elsewhere, I have demonstrated my own extension of this bracket, allowing for subscripts: [a, b]_ij º a_ib_j — b_ia_j.
Please note that terms inside the bracket permute, but subscripts do not. Then, we can write the above DETERMINANT as: [a, b]_ij. But what the numerator terms in each solution? Note that this (the "secondary determinant") derives from the denominator by replacing constant terms for the coefficient of the "unknown" being solved: c_i for the a_i-coefficients of x in the x-solution; c_i for the b_i-coefficients of y in the y-solution -- respectively, [b, c]₁₂ and [a, c]₁₂.
Then, we can succinctly write: x = [b, c]₁₂/[a, b]₁₂; y = [a, c]₁₂/[a, b]₁₂.
However, the following, rather than my bracket-form, the conventional notation for the (primary) determinant:
|a₁ b₁| |a₂ b₂| = a₁b₂ — a₂b₁
And the other forms are adapted, accordingly.
Please note that the determinantal form (as above) is "square": EQUAL NUMBER OF ROWS AND COLUMNS. British mathematician, Arthur Cayley (1821-1895), then GENERALIZED TO THE MATRIX, WITH POSSIBLY DIFFERENT NUMBER ROWS AND COLUMNS. Using the fact that "a determinant is a number", and we know how to multiply numbers, Cayley derived a multiplication-form for matrices -- "row-terms of left matrix multiply column-terms of right matrix". Etc.
RETURN TO OUTERPRODUCT
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