The determinant function can be defined by essentially two different methods. The advantage of the first definition—one which uses permutations—is that it provides an actual formula for det A, a fact of theoretical importance. The disadvantage is that, quite frankly, no one actually computes a determinant by this method.
Method 1 for defining the determinant. If n is a positive integer, then a permutation of the set S = {1, 2, …, n} is defined to be a bijective function—that is, a one-to-one correspondence—σ, from S to S. For example, let S = {1, 2, 3} and define a permutation σ of S as follows:
Since σ(1) = 3, σ(2) = 1, and σ(3) = 2, the permutation σ maps the elements 1, 2, 3 into 3, 1, 2. Intuitively, then, a permutation of the set S = {1, 2, …, n} provides a rearrangement of the numbers 1, 2, …, n. Another permutation, σ′, of the set S is defined as follows:
This permutation maps the elements 1, 2, 3 into 2, 1, 3, respectively. This result is written
Hope the above explanation was useful, not let me explain you about adjoint matrix.
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