Cofactor Expansion: Cramer s Rule

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1 Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015

2 Introduction Today we will focus on developing: an efficient method for calculating determinants of n n matrices, a method for finding the inverse of an invertible matrix, and a formula for finding the solution to certain linear systems in terms of determinants.

3 General 3 3 Determinant Note: a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 13 a 22 a 31 a 12 a 21 a 33 a 11 a 23 a 32 = a 11 (a 22 a 33 a 23 a 32 ) a 12 (a 21 a 33 a 23 a 31 ) + a 13 (a 21 a 32 a 22 a 31 ) a = a 22 a a 32 a 33 a 12 a 21 a 23 a 31 a 33 a + a 21 a a 31 a 32 = a 11 M 11 a 12 M 12 + a 13 M 13

4 Minors and Cofactors Definition If A is a square matrix, then the minor of entry a ij is denoted M ij and is defined to be the determinant of the submatrix that remains after the i th row and j th column are deleted from A. The number ( 1) i+j M ij is denoted by C ij and is called the cofactor of entry a ij. For a 3 3 matrix A, det(a) = a 11 M 11 a 12 M 12 +a 13 M 13 = a 11 C 11 +a 12 C 12 +a 13 C 13. This is called the cofactor expansion along the first row of A.

5 Minors and Cofactors Definition If A is a square matrix, then the minor of entry a ij is denoted M ij and is defined to be the determinant of the submatrix that remains after the i th row and j th column are deleted from A. The number ( 1) i+j M ij is denoted by C ij and is called the cofactor of entry a ij. For a 3 3 matrix A, det(a) = a 11 M 11 a 12 M 12 +a 13 M 13 = a 11 C 11 +a 12 C 12 +a 13 C 13. This is called the cofactor expansion along the first row of A.

6 Example Example Let A = cofactors. 1. M M C C and find the following minors and

7 Cofactor Expansions Theorem The determinant of an n n matrix A can be computed by multiplying the entries in any row (column) by their corresponding cofactors and adding the resulting products. det(a) = a i1 C i1 + a i2 C i2 + + a in C } {{ in } cofactor expansion along i th row = a 1j C 1j + a 2j C 2j + + a nj C nj } {{ } cofactor expansion along j th column Strategy: expand along the row or column containing the most zeros.

8 Cofactor Expansions Theorem The determinant of an n n matrix A can be computed by multiplying the entries in any row (column) by their corresponding cofactors and adding the resulting products. det(a) = a i1 C i1 + a i2 C i2 + + a in C } {{ in } cofactor expansion along i th row = a 1j C 1j + a 2j C 2j + + a nj C nj } {{ } cofactor expansion along j th column Strategy: expand along the row or column containing the most zeros.

9 Example Compute the following determinant

10 Adjoint of a Matrix Observation: if we multiply the entries in a row (column) of a matrix by the corresponding cofactors from a different row (column), the sum of these products will be 0. Example Let A = and find the following sum. a 11 C 21 + a 12 C 22 + a 13 C 23

11 Adjoint of a Matrix Observation: if we multiply the entries in a row (column) of a matrix by the corresponding cofactors from a different row (column), the sum of these products will be 0. Example Let A = and find the following sum. a 11 C 21 + a 12 C 22 + a 13 C 23

12 Adjoint of a Matrix (continued) Definition If A is an n n matrix and C ij is the cofactor of a ij, then the matrix C 11 C 12 C 1n C 21 C 22 C 2n... C n1 C n2 C nn is called the matrix of cofactors from A. The transpose of this matrix is called the adjoint of A and is denoted adj(a).

13 Example Example Let A = adj(a) and find the matrix of cofactors and

14 Inverses and Adjoints Theorem If A is an invertible matrix, then A 1 = 1 det(a) adj(a). Proof. a 11 a 12 a 1n a 21 a 22 a 2n... a i1 a i2 a in... a n1 a n2 a nn C 11 C 21 C j1 C n1 C 12 C 22 C j2 C n2.... C 1n C 2n C jn C nn

15 Inverses and Adjoints Theorem If A is an invertible matrix, then A 1 = 1 det(a) adj(a). Proof. a 11 a 12 a 1n a 21 a 22 a 2n... a i1 a i2 a in... a n1 a n2 a nn C 11 C 21 C j1 C n1 C 12 C 22 C j2 C n2.... C 1n C 2n C jn C nn

16 Example Example Let A = A and use the formula 1 adj(a) to find det(a)

17 Cramer s Rule Cramer s Rule is a theoretical method for determining the solution to a linear system. It is computationally inefficient. Theorem If Ax = b is a system of n linear equations in n unknowns such that det(a) 0, then the system has a unique solution: x 1 = det(a 1) det(a), x 2 = det(a 2) det(a),..., x n = det(a n) det(a). where A j is the matrix obtained when the entries in the j th column of A are replaced by b. Proof.

18 Cramer s Rule Cramer s Rule is a theoretical method for determining the solution to a linear system. It is computationally inefficient. Theorem If Ax = b is a system of n linear equations in n unknowns such that det(a) 0, then the system has a unique solution: x 1 = det(a 1) det(a), x 2 = det(a 2) det(a),..., x n = det(a n) det(a). where A j is the matrix obtained when the entries in the j th column of A are replaced by b. Proof.

19 Cramer s Rule Cramer s Rule is a theoretical method for determining the solution to a linear system. It is computationally inefficient. Theorem If Ax = b is a system of n linear equations in n unknowns such that det(a) 0, then the system has a unique solution: x 1 = det(a 1) det(a), x 2 = det(a 2) det(a),..., x n = det(a n) det(a). where A j is the matrix obtained when the entries in the j th column of A are replaced by b. Proof.

20 Example Example Use Cramer s Rule to solve the following linear system. 3x 1 + 3x 2 + x 3 = 1 x 1 4x 3 = 1 x 1 3x 2 + 5x 3 = 1 Remark: for linear systems with more than 3 equations and 3 unknowns, Gaussian elimination is more efficient for finding the solution than Cramer s Rule.

21 Example Example Use Cramer s Rule to solve the following linear system. 3x 1 + 3x 2 + x 3 = 1 x 1 4x 3 = 1 x 1 3x 2 + 5x 3 = 1 Remark: for linear systems with more than 3 equations and 3 unknowns, Gaussian elimination is more efficient for finding the solution than Cramer s Rule.

22 Homework Read Section 2.4 and work exercises 1, 3, 4, 5, 7, 9, 13, 17, 25, 26, 27.

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