�Determinants�

• Properties of determinants
• If the elements of a row (column) are scaled by a constant, then the determinant will also be scaled by that constant.
• Interchanging two rows or columns changes the sign of the determinant.
• Read on other properties.
• Matrix representation
• Any system of linear equations of the form
• a₁₁x₁+a₁₂x₂+…..+a_1nx_n = b₁
• a₂₁x₁+a₂₂x₂+…..+a_2nx_n = b₂
• _{…………………………………………}
• a_m1x₁+a_m2x₂+…..+a_mnx_n = b_m
• In matrix notation, we have:
• AX = B, where A is the coefficient matrix.

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�The Gaussian Method�

• To use the Gaussian elimination method of solving a system of linear equations, simply express the system of equations as an augmented matrix [A | B] and apply row operations to the augmented matrix until the original coefficient matrix to the left of the bar is reduced to an identity matrix.
• The advantages of this method are that we don’t check whether the determinant is non-zero before proceeding, and it will handle rectangular system of equations.

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• Given AX = B …. (1) , if A is a square matrix and non-singular, then A^-1 exist.
• Pre-multiplying (1) by A^-1 gives us
• A^-1(AX) = A^-1B
• (A^-1A)X = A^-1B (Associative law)
• IX = A^-1B
• ∴ X = A^-1B
• Hence, if we find A^-1, we can then post- multiply it by B to get the matrix X.
• There are 2 alternative methods for solving a system of equations which do not directly involve finding the inverse. There are the Cramer’s rule and the Gaussian method.

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• Trial question
• Use the Cramer’s rule to solve for x₁, x₂ and x₃, given the system of linear equations:
•  
• 4x₁ + 2x₂ + 5x₃ = 21
• 3x₁ + 6x₂ + x₃ = 31
• x₁ + 8x₂ + 3x₃ = 37
•  
• 4x₁ + 2x₂ + 7x₃ = 35
• 3x₁ + x₂ + 8x₃ = 25
• 5 x₁+ 3x₂ + x₃ = 40
•  

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• Example  
• x₁ + 3x₂ = 5
• 2x₁ + 6x₂ = 11
• has a determinant zero. These are 2 parallel lines, therefore the system is INCONSISTENT. There is no solution.
• Also 3x₁ + 2x₂ + x₃ = 10
• 2x₁ + 3x₂ - x₃ = 5
• 4 x₁+ x₂+3x₃ = 15
• has a determinant of zero. The last row is a linear combination of the other two since R3 = 2R1 – R2. The system is CONSISTENT, but INDETERMINATE. There is no unique solution

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�Applications �

• Example 2
• Use Cramer’s rule to solve for P* and Q* in each of the following three interconnected markets:
• Q_s1 = 6P₁ − 8 ; Q_d1 = −5P₁ + P₂ + P₃ + 23
• Q_s2 = 3P₂ − 11 ; Q_d2 = P₁ − 3P₂ + 2P₃ + 15
• Q_s3 = 3P₃ − 5 ; Q_d3 = P₁ + 2P₂ − 4P₃ + 19
•  
• Soln
• A market is in equilibrium when Q_d = Q_s. The markets are simultaneously in equilibrium when
• 11P₁ − P₂ − P₃ = 31
• −P₁ + 6P₂ − 2P₃ = 26
• −P₁ − 2P₂ + 7P₃ = 24

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