Determinants
Introduction:
If A is a square matrix then the determinant function is denoted by det and det(A)
For an instance we have a 2 x 2 matrix denominated A, therefore:
det(A) = [a11 , a12 ; a21 , a22 ]
As we already defined the determinant function we can write some formulas. The formulas for any 2 x 2 and 3 x 3 matrix will be:
The determinant function for a 2 x 2 matrix is:
$ det(A)=\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right) $
= (a11 * a22) - (a12 * a21 )
The determinant function for a 3 x 3 matrix is:
$ det(A)=\left(\begin{array}{cccc}a11&a12&a13\\a21&a22&a23\\a31&a32&a33\end{array}\right) $
= (a11 * a22 * a33) + (a12 * a23 * a31) + (a13 * a21 * a32) - (a12 * a21 * a33) - (a11 * a23 * a32) - (a13 * a22 * a31)
Properties of Determinants:
Theorem 1: Let A be an n x n matrix then; det(A) = det(At)
Theorem 2: If a matrix B results from matrix A by interchanging two different rows (columns) of A, then; det(B) = - det(A)
Theorem 3: If two rows (columns) of A are equal, then; det(A) = 0
Theorem 4: If a row (column) of A consists entirely of zeros, then; det(A) = 0
Theorem 5: If B obtained from A by multiplying a row (column) of A by a real number k, then;det(B) = kdet(A)
Theorem 6: If B = [bij] is obained from A = [aij] by adding to each element of the rth row (column) of A, k times the corresponding element of the sth row (column), r not equal s, of A, then; det(B) = det(A)
Theorem 7: If a matrix A = [aij] is upper (lower) triangular, then; det(A) = a11*a12...ann ; tha is, the determinant of a triangular matrix is the product of the element on themain diagonal.
Theorem 8: If A is an n x n matrix, then A is nonsingular if and only if det(A) not equal 0
Theorem 9: If A and B are n x n matrices, then; det(AB) = det(A)det(B)
Cofactor Expansion:
The cofactor expansion is a method for evaluating the determinant of an n xn matrix that reduces the problem to the evaluation of determinants of matrices of order n - 1. We should repeat the proces of (n-1) x (n-1) until we have a 2 x 2 matrices.
Let A = [aij] be an n x n matrix. Let Mij be the (n-1) x (n-1) submatrix of A obtained by deleting the ith row and jth row column of A. The determinant det(Mij) is called the minor aij. Also, Let A = [aij] be an n x n matrix. The cofactor Aij of aij is defined as Aij = (-1)i+j det(Mij)
Theorem 10: Let A = [aij] be an n x n matrix. then;
det(A) = ai1Ai1+ai2Ai2+...+ainAin and det(A)=a1jA1j+a2jA2j+...+anjAnj
[expansion of det(A) along the ith row] [expansion of det(A) along the jth column]
Inverse of a Matrix:
Theorem 11:
