3 5 Solution by Determinants The Determinant of

3. 5 Solution by Determinants

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

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

Coefficient Matrix n n n You can use determinants to solve a system of linear equations You use the coefficient matrix of the linear system Linear System Coeff Matrix ax+by = e cx+dy = f

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

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

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

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

Example 2 - Cramer’s Rule (3 x 3) n 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)!!!

Inverse Matrix

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

Solving Equations Using Inverse Matrices n 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…

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

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

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

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

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: Unless a = 0. Does matrix multiplication have an identity? Yes! Unless det(A) = 0.