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| − | <strike></strike><sub></sub>Theorem 12: If A = [a<sub>ij</sub>] is an n x n matrix, then; '''A(adj A) = (adj A)A = det(A)I<sub>n</sub>.''' | + | <strike></strike><sub></sub><u>Theorem 12:</u> If A = [a<sub>ij</sub>] is an n x n matrix, then; '''A(adj A) = (adj A)A = det(A)I<sub>n</sub>.''' |
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| − | Theorem 13: Cramer's Rule | + | <u>Theorem 13:</u> Cramer's Rule |
Let; | Let; | ||
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| − | We note that Cramer's rule is only applicable when we have n equations in n unknowns and the coefficient matrix A is nonsingular. | + | We note that Cramer's rule is only applicable when we have n equations in n unknowns and the coefficient matrix A is nonsingular. If we are facing a linear system of n equations in n unknowns whose coefficient matrix is singular, we must use the Gaussian elimination or Gauss-Jordan reduction methods. |
| + | |||
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| + | <u>At this point of learning we have shown the following:</u> | ||
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| + | 1. A is nonsingular | ||
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| + | 2. A'''x = 0''' has only the trival solution | ||
| + | |||
| + | 3. A is row (column) equivalent to In | ||
| + | |||
| + | 4. The linear system A'''x = b''' has a unique solution for every n x 1 matrix '''b'''. | ||
| + | |||
| + | 5. A is a product of elementray matrices | ||
| + | |||
| + | 6. det(A) not equal 0 | ||
Revision as of 17:33, 7 December 2011
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)=\left(\begin{array}{cccc}a11&a12\\a21&a22\end{array}\right) $
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: If A = [aij] is an n x nmatrix, then;
ai1Akl+ai2Ak2+...+ainAkn = 0 for i not equal k ; a1jA1k+a2jA2k+...+anjAnk 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 = [aij] is an n x n matrix, then; A(adj A) = (adj A)A = det(A)In.
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 + ... + a1nxn = b1
a21x1 + a22x2 + ... + a2nxn = b2
...
an1x1 + an2x2 + ... + annxn = bn
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), ..., xn = det(An)/det(A),
where Ai is the matrix obtained from A by replacing the ith column of A by b.
We note that Cramer's rule is only applicable when we have n equations in n unknowns and the coefficient matrix A is nonsingular. If we are facing a linear system of n equations in n unknowns whose coefficient matrix is singular, we must use the Gaussian elimination or Gauss-Jordan reduction methods.
At this point of learning we have shown the following:
1. A is nonsingular
2. Ax = 0 has only the trival solution
3. A is row (column) equivalent to In
4. The linear system Ax = b has a unique solution for every n x 1 matrix b.
5. A is a product of elementray matrices
6. det(A) not equal 0
