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3.5 Solution by Determinants

3.5 Solution by Determinants. The Determinant of a Matrix. The determinant of a matrix A is denoted by |A|. Determinants exist only for square matrices. The Determinant for a 2x2 matrix. If A = Then This one is easy. Coefficient Matrix.

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3.5 Solution by Determinants

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  1. 3.5 Solution by Determinants

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

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

  4. 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 ax+by = e cx+dy = f

  5. Cramer’s Rule • Linear SystemCoeff Matrix ax+by = e cx+dy = f • Let D be the coefficient matrix • If det D≠ 0, then the system has exactly one solution: and

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

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

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

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

  10. Inverse Matrix

  11. 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. 2x + 3y – 2z = –2 –4x + 2y + 3z = 1 5x + 7y + 6z = 28 Then the system can be written as:

  12. 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…

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

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

  15. 2 -5 -1 3 1 0 0 1 2 -5 -1 3 3 5 1 2 A = AB = = = I How to find the Inverse Matrix (cont’d) 3 5 1 2 Is the inverse of B =A-1 = 3 5 1 2 2 -5 -1 3 1 0 0 1 BA = = = I

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

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

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