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ss_10_4 - Section 10.4 Determinants Determinants are...

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Section 10.4 Determinants Determinants are important mathematical tools that can frequently be used to analyze and solve linear systems. There are various methods for finding determinants. Some of the simpler methods only apply to 2 x 2 and 3 x 3 matrices. Determinants of 2 x 2 matrices If D = a b c d is a 2 by 2 matrix, then the determinant of D is det ( D ) = a b c d = ad - bc Example. Find the determinant of D = 1 - 1 2 - 3 Solution. 1 - 1 2 - 3 = (1)( - 3) - ( - 1)(2) = - 1 Determinants of 3 x 3 matrices If D = a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 is a 3 x 3 matrix, then the determinant of D is det ( D ) = 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 ) The following is a good device for remembering how to evaluate 3 x 3 determinants. Example. Find the determinant of D = 1 - 1 0 2 1 1 - 1 - 2 - 1 Solution. 1 - 1 0 2 1 1 - 1 - 2 - 1 = (1)(1)( - 1) + ( - 1)(1)( - 1) + (0)(2)( - 2) - ((0)(1)( - 1) + ( - 1)(2)( - 1) + (1)(1)( - 2)) 1

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= - 1 + 1 + 0 - (0 + 2 - 2) = 0 Important note. The methods shown above for 2 x 2 and 3 x 3 determinants does NOT apply to 4 x 4 or higher-order determinants. Cramer’s Rule Cramer’s rule is a method that can be used for solving linear systems when the number of unknowns is the same as the number of equations and the determinant of the system’s coefficient matrix is not equal to zero.
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ss_10_4 - Section 10.4 Determinants Determinants are...

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