Chapter 6.2, Problem 55E

For a fixed positive integer n, let D be a function whichassigns to any $n\times n$ matrix A a number D(A) such that
a. D is linear in the rows (see Theorem 6.2.2),
b. $D\left(B\right)=-D\left(A\right)$ if B is obtained from A by a rowswap, and
c. $D\left(I_n\right)=1$ .
Show that $D\left(A\right)=\det \left(A\right)$ for all $n\times n$ matrices A.Him: Consider $E=\text{rref}A$ . Think about the relationship between D(A) and D(E), mimicking Algorithm 6.2.5.The point of this exercise is that det(A) can be characterized by the three properties a, b. and c; the determinant can, in fact, be defined in terms of these properties. Ever since this approach was first presented in the1 880s by the German mathematician Karl Weierstrass (1817-1897), this definition has been generally use din advanced linear algebra courses because it allows amore elegant presentation of the theory of determinants.