What a determinant really determines

The determinant is an important concept in linear algebra. Since a determinant is defined for a square matrix, many people would think it as a property of a matrix. But what does it really determine? In other words, why did mathematicians invent it? And why is it defined so? For example, why $\ \det\left ( \left [ \array{ a & b \cr c & d} \right ] \right) = ad - bc $?

Answer: It determines whether a system of linear equations has a unique solution.

The concept of matrix does not come from nowhere. It is strongly related to linear equations. Let's consider a system of linear equations: $$ \begin{align}
ax + by & = C_1\\
cx + dy & = C_2
\label{eq:two_eqs}
\end{align} $$

If $C_1=C_2=0$, eliminating $x$ will result in this: $(ad-bc)y=0$. Pay attention to the coefficient: $ad-bc$. Does it look like $\ \det\left ( \left [ \array{ a & b \cr c & d} \right ] \right) $?

Now if $ad-bc=0$, then $y$ can take whatever value to be a solution of the equations. Namely, the equation has unlimited amount of solutions.

If $C_1 \not = 0$ and $C_2 \not = 0$, the result of eliminating $x$ will have a non-zero constant on the right-hand side: $(ad-bc)y=C_3$. Now if $ad-bc = 0$, there is no way for the equations to have a solution.

Therefore, the determinant actually determines whether a system of linear equations has a unique solution. A system of linear equations can be represented as a matrix. So the determinant of the matrix defines the property of the linear system that the set of equations defines.

References:
1. System of Linear Equations, http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/system/system.html
2. Determinant, Wolfram MathWorld, http://mathworld.wolfram.com/Determinant.html
3. Determinant, Wikipedia, http://en.wikipedia.org/wiki/Determinant

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