The name 'determinant' implies that it determines. It determines the value of variable x3 by calculating its coefficient in the linear system of equations say, a1x1 + a2x2 + a3x3 = c1; a4x1 + a5x2 + a6x3 = c2; a7x1 + a8x2 + a9x3 = c3; using Gauss-elimination, where a1 through a9 are constants, x1, x2 and x3 are the variables whose values will give the solution to the system of equations, and c1, c2 and c3 are also constants though they form no part of the determinant. The determinant is represented as | (first row) a1 a2 a3 (second row) a4 a5 a6 (third row) a7 a8 a9 |. This is how I understand the motivation of determinants. In all textbooks however they give the standard formula without explaining the motivation, viz., that a1(a5.a9 - a8.a6) - a2(a4.a9 - a7.a6) + a3(a4.a8 - a7.a5) = 0 derives from cross-multiplication of the two-variate system a1x1 + a2x2 = a3; a4x1 + a5x2 = a6; a7x1 + a8x2 = a9. I don't see the similarity of steps or congruence between this and the Gauss-elimination method that I understand as the real motivation and that I stated first, though both methods give the same result. Could anyone illumine?