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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(A^t)

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 = [b_ij] is obained from A = [a_ij] 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 = [a_ij] is upper (lower) triangular, then; det(A) = a₁₁*a₁₂...a_nn ; 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 = [a_ij] 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(M_ij) is called the minor a_ij. Also, Let A = [a_ij] be an n x n matrix. The cofactor A_ij of a_ij is defined as A_ij = (-1)^i+j det(M_ij)

Theorem 10: Let A = [aij] be an n x n matrix. then;

                                det(A) = a_i1A_i1+a_i2A_i2+...+a_inA_in                             and                        det(A)=a_1jA_1j+a_2jA_2j+...+a_njA_nj

                            [expansion of det(A) along the ith row]                                                [expansion of det(A) along the jth column]

Inverse of a Matrix:

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Theorem 11: If A = [aij] is an n x nmatrix, then; 

                                      a_i1A_kl+a_i2A_k2+...+a_inA_kn = 0    for i not equal k    ;    a_1jA_1k+a_2jA_2k+...+a_njA_nk    for j not equal k

Let A = [aij] be an n x n matrix. Then n xn adj A, called the adjoint of A, is the matrix whose (i,j)th entry is the cofactor Aji of aji. Thus;

                                                                     $ adj A=\left(\begin{array}{cccc}A11&A21&...&An1\\A12&A22&...&An2\\...&...&...&...\\A1n&A2n&...&Ann\end{array}\right) $

Theorem 12: If A = [a_ij] is an n x n matrix, then; A(adj A) = (adj A)A = det(A)I_n.

Other applications of Determinants:

To obtain another method for solving a linear system of n equations in n unknowns is known as the Cramer's Rule.

Theorem 13: Cramer's Rule

                                                                        Let;

                                                                                           a₁₁x₁ + a₁₂x₂ + ... + a_1nx_n = b₁

                                                                                           a₂₁x₁ + a₂₂x₂ + ... + a_2nx_n = b₂

                                                                                                                   ... 

                                                                                           a_n1x₁ + a_n2x₂ + ... + a_nnx_n = b_n

be a linear system of n equations in n unknowns, and let A = [aij] be the coefficient matrix so that we can write the given system as Ax = b, where

                                                                                                    $ b=\left(\begin{array}{cccc}b1\\b2\\...\\bn\end{array}\right) $

If det(A) not equal 0, then the system has the unique solutions

                                                               x₁ = det(A₁)/det(A),      x₂ = det(A₂)/det(A),      ...,      x_n = det(A_n)/det(A),

where Ai is the matrix obtained from A by replacing the ith column of A by b.