8
Feb 19

Determinants: starting simple




Determinants: starting simple

Previously  we looked at a motivating example to consider the determinant \det A=ad-bc of a 2\times 2 matrix

A=\left(\begin{array}{cc}a&b\\c&d\end{array}\right).

Using this basic example, now we formulate properties that uniquely define determinants of matrices of higher orders. The discussion is based mainly on Kurosh, Course in linear algebra, 9th edition, Moscow, 1968 (in Russian).

Observation 1. Homogeneity. If one of the rows of A is multiplied by a number k, then \det A gets multiplied by k,

\det \left(\begin{array}{cc}ka&kb\\c&d\end{array}\right)    =kad-kbc=k\det A, \det \left(\begin{array}{cc}a&b\\kc&kd\end{array}\right)=akd-bkc=k\det A.

Observation 2. Adding one of the rows of A to another doesn't change the value of the determinant:

\det \left(\begin{array}{cc}a+c&b+d\\c&d\end{array}\right)=(a+c)d-(b+d)c=\det A, \det \left(\begin{array}{cc}a&b\\a+c&b+d\end{array}\right)=a(b+d)-b(a+c)=\det A.

To see the intuition behind these rules, recall that the purpose of the determinant is to verify whether the system Ax=y has solutions. Homogeneity means that if one of the equations of the system is multiplied by a nonzero constant, the solvability of the new system will be equivalent to the solvability of the original system. Similarly, if one of the equations of the system is added to another, solvability of the system as judged by the determinant will not change. This makes sense because multiplying a system equation by a nonzero constant or adding one equation to another does not change the information contained in the system.

Keep in mind an emerging general idea: the determinant discards any transformations of the matrix that do not impact its invertibility.

Observation 3. Determinant of the identity: \det I=1.

Taking these properties as axioms for matrices of higher order, we will show that they uniquely define determinants and develop a couple of rules involving determinants. One of them is the Leibniz formula for determinants and the other is Cramer's rule.

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