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3.5 Solution by Determinants

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The Determinant of a Matrix The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices.

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The Determinant for a 2x2 matrix If A = Then This one is easy

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Coefficient Matrix You can use determinants to solve a system of linear equations You use the coefficient matrix of the linear system Linear SystemCoeff Matrix a x +b y = e c x +d y = f

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Cramer’s Rule Linear SystemCoeff Matrix a x +b y = e c x +d y = f Let D be the coefficient matrix If det D ≠ 0, then the system has exactly one solution: and

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Example 1- Cramer’s Rule (2x2) Solve the system: 8 x + 5 y = 2 2 x ─ 4 y = −10 The coefficient matrix is: and So: and

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Solution: (-1,2) Example 1 (continued)

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Value of 3 x 3 (4 x 4, 5 x 5, etc.) determinants can be found using so called expansion by minors. The Determinant for a 3x3 matrix

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Example 2 - Cramer’s Rule (3x3) Solve the system: x + 3 y – z = 1 –2 x – 6 y + z = –3 3 x + 5 y – 2 z = 4 Let’s solve for Z The answer is: (2,0,1)!!!

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Inverse Matrix

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Using Matrix-Matrix Multiplication: 2x + 3y – 2z –4x + 2y + 3z 5x + 7y + 6z This gives us a simple way to write a system of linear equations. Then the system 2x + 3y – 2z = –2 –4x + 2y + 3z = 1 5x + 7y + 6z = 28 can be written as:

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Solving Equations Using Inverse Matrices If A is the matrix of coefficients, X is the matrix of variables and B is the matrix of constants, then a system of equations can be presented as a matrix equation…

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…and we can solve it for X by multiplying both sides of the equation by A -1 from the left:

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How to find the Inverse Matrix For a 2x2 matrix: a b c d A = If ad – bc ≠ 0 then: d -b -c a A -1 = 1 ad – bc d ad-bc -b ad-bc -c ad-bc a ad-bc =

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3 5 1 2 B =A -1 = 2 -5 -1 3 A = Is the inverse of 2 -5 -1 3 AB = 3 5 1 2 = 1 0 0 1 = I 2 -5 -1 3 BA = 3 5 1 2 = 1 0 0 1 = I How to find the Inverse Matrix (cont’d)

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Find the inverse of 1 2 1 3 A =A = d -b -c a 1 ad-bc d ad-bc -b ad-bc -c ad-bc a ad-bc = Using the formula: a=1; b=2; c=1; d=3 Since ad – bc = 3 – 2=1: d -b -c a = 3 -2 -1 1

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Properties Real-number multiplication is commutative: Is matrix multiplication commutative? No! Real-number multiplication is associative: Is matrix multiplication associative? Yes! Real-number multiplication has an identity: Does matrix multiplication have an identity? Yes! (but you must use an identity matrix of the proper size for A) Real-number multiplication has inverses: Does matrix multiplication have an identity? Unless a = 0. Yes! Unless det( A) = 0.

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