1050 Section 8.3 Notes(answers).pdf - Math 1050 Section 8.3 Solving Systems using Determinants(Cramer’s Rule Solving systems using row operations is a

1050 Section 8.3 Notes(answers).pdf - Math 1050 Section 8.3...

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Math 1050 - Section 8.3 - Solving Systems using Determinants (Cramer’s Rule) Solving systems using row operations is a sure way of getting the answers , especially if the system is inconsistent or dependent. However, there are other ways of getting the answers that are easier…when they work. Determinant- A unique number associated with each ___ square __ matrix that contains information about the numbers in the matrix and their position. This is a determinant: | 3 1 4 2 | This is a matrix: [ 3 1 4 2 ] The determinant of | ? ? ? ? | is ?? − ?? . This can be found by multiplying the diagonals and subtracting. Example 1- a. | 3 1 4 2 | b. | −6 1 −3 2 | c. | −6 1 −3 2 4 0 | Determinant of a 3x3 : Each position in a matrix has a sign associated with it. The sign has ___ nothing __ to do with the sign of the number in that position. The sign is determined solely by the _ position __. The signs in a matrix are easy to remember because they make a “checkerboard” of + ’s and ‘s as follows: Example 2- Find the determinant 1 2 3 3 3 1 4 5 6 = 3 1 2 1 2 3 1 2 3 5 6 4 6 4 5  
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Unformatted text preview: Example 3-Find the determinant | 1 1 1 1 −5 1 −2 3 | | + − + − + − + − + ⋯ ⋯ ⋯ ⋮ ⋮ ⋮ ⋱ | Example 4- Find ? if | ? 1 3 ? | = −2 . Cramer’s Rule- Cramer’s Rule allows us to solve systems using _ determinants __. To solve the 2x2 system { ? ? + ? − ? = = 8 4 we need to compute three different determinants: - the determinant of the matrix of coefficients ?- the determinant of the ? matrix (i.e. the ? column gets replaced by the _ constant _ column.) ?- the determinant of the ? matrix (i.e. the ? column gets replaced by the _ constant __ column.) Once we’ve found those, Cramer’s Rule says th at: ? = ? and ? = ? Example 3- Solve using Cramer’s Rule . a. ? ? + ? − ? = = 8 4 b. ? 3? −2? −5? = = 2 9 If Cramer’s Rule says that ? = ? and ? = ? for a 2x2 system, what do you think the rule is for a 3x3 system? ? = ? and ? = ? and ? = ?...
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