DETERMINANTS November 2014 Determinants If a matrix is
DETERMINANTS November 2014
Determinants • If a matrix is square (that is, if it has the same number of rows as columns), then we can assign to it a number called its determinant. § Determinants can be used to solve systems of linear equations. § They are also useful in determining whether a matrix has an inverse.
Determinants… • Notation: • We denote the determinant of a square matrix A by the symbol det (A) or | A |. • We can also designate the determinant of matrix A by replacing the brackets by vertical straight lines. For example
Determinant of 1 x 1 matrix • We first define det(A) for the simplest cases. § If A = [a] is a 1 x 1 matrix, then det(A) = a. Definition: The determinant of a 1 1 matrix [a] is the scalar a. Example: The determinant of the matrix [5] is 5 and the determinant of the matrix [-0. 43] is -0. 43
Determinant of a 2 x 2 Matrix • Definition: The determinant of a 2 2 matrix is the scalar a 11 a 22 -a 12 a 21 i. e. ,
Determinant of a 2 x 2 Matrix… • Example : Evaluate the determinant of • Solution:
Determinant of an n × n Matrix • To define the concept of determinant for an arbitrary n × n matrix, we need the following terminology. • Let A=[aij] be an n × n matrix. § The minor Mij of the element aij is the determinant of the (n – 1) × (n – 1) submatrix that remains after the ith row and jth column are deleted from A. § The cofactor Cij of the element aij is: Cij = (– 1)i + j. Mij
Determinant of an n × n Matrix… • For example • Note that the cofactor of aij is simply the minor of aij multiplied by either 1 or – 1, depending on whether i + j is even or odd.
Determinant of an n × n Matrix… • Example: Let • The minor of entry a 11 is • The cofactor of a 11 is
Determinant of an n × n Matrix… • Similarly, the minor of entry a 32 is • The cofactor of a 32 is
Determinant of an n × n Matrix… • To find the determinant of a matrix A of arbitrary order, a) Pick any one row or any one column of the matrix; b)For each element in the row or column chosen, find its cofactor; c) Multiply each element in the row or column chosen by its cofactor and sum the results. This sum is the determinant of the matrix.
• In other words, the determinant of A is given by 1 st row expansion ith row expansion jth column expansion
Example : We can compute the determinant by expanding along the first row,
Or expand down the second column: Or using a row or column with many zeros:
Evaluating Determinants By Row Reduction • Theorem 1 Let A be a square matrix. § If A has a row of zeros or a column of zeros, then det A = 0. § det (A) = det (AT ). • Theorem 2 (Triangular Matrix): If A is an n×n triangular matrix (upper triangular, lower triangular, or diagonal), then det(A) is the product of the entries on the main diagonal of the matrix ; that is, det(A) = a 11 a 22. . . ann
• Theorem 3 (Elementary Row Operations) Let A be an n × n matrix. • If B is the matrix that results when a single row or single column of A is multiplied by a scalar k, then det (A) = k det (B) • If B is the matrix that results when two rows or two columns of A are interchanged, then det (A) = -det (B) • If B is the matrix that results when a multiple of one row of A is added to another row or when a multiple of one column of A is added to another column, then det (A) = det (B).
Example: Consider the matrices If we evaluate the determinants of these matrices we obtain det (A) = -2, det (A 1) = -8, det (A 2) = 2 and det (A 3)=-2
Observe that A 1 is obtained by multiplying the first row of A by 4; A 2 by interchanging the first two rows; and A 3 by adding -2 times the first row of A to the second. As predicted by Theorem 3 we have the relationships; det (A 1) = 4 det (A) det (A 2) = -det (A) and det (A 3)= det (A)
Example (Using Row Reduction to Evaluate a Determinant) • Evaluate det(A) where • Solution: The first and second rows of A are interchanged. A common factor of 3 from the first row was taken through the determinant sign
-2 times the first row was added to the third row. -10 times the second row was added to the third row A common factor of 55 from the last row was taken through the determinant sign.
• Example: • Solution: We have
Theorem 4 (Matrices with Proportional Rows or Columns) • If A is a square matrix with two proportional rows or two proportional column, then det(A) = 0. Example: (Introducing Zero Rows) • Evaluate det(A) where
• Solution: The second row is 2 times the first, so we added -2 times the first row to the second to introduce a row of zeros.
Theorem 5: For an n n matrix A and any scalar k, Theorem 6: If A and B are square matrices of the same size, then
Adjoint of a Matrix • Definition: If A is any n × n matrix and Cij is the cofactor of aij, then the matrix is called the matrix of cofactors from A. • The transpose of this matrix is called the adjoint of A and is denoted by adj(A)
Example of finding adjoint • Find the adjoint of Solution: The cofactors matrix of A are:
The matrix of cofactors is
Theorem 7: A square matrix has an inverse (is invertible) if and only if its determinant is not zero. Corollary: If A is invertible, then Below we develop a method to calculate the inverse of nonsingular matrices using determinants.
Inverse of a Matrix Using Its Adjoint Theorem 8: If A is an n × n matrix, then A [adj(A)] = [adj(A)] A = |A| In. If |A| ≠ 0 then from the Theorem above,
Theorem 9 (Inverse of a Matrix Using Its Adjoint) If A is an n × n matrix and det A ≠ 0, then That is, if |A| ≠ 0, then A-1 may be obtained by dividing the adjoint of A by the determinant of A. Example: Let
Inverse of a Matrix… • The cofactors of A are • so that the matrix of cofactors is
Inverse of a Matrix… • and the adjoint of A is • the determinant of A is
Inverse of a Matrix… • Thus
The matrix inverse can be computed as follows 1. Find the determinant det A 2. Find the cofactors of all elements in A and form a new matrix C of cofactors, where each element is replaced by its cofactor. 3. The inverse of A is now given as Note: the inverse A− 1 exists if (and only if) det A ≠ 0.
Example: Use the adjoint of The cofactors of A are So the matrix of cofactors is to find A-1
and the adjoint of A is
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