Matrices and Determinants

Transcription

1 Math Assignment Eperts is a leading provider of online Math help. Our eperts have prepared sample assignments to demonstrate the quality of solution we provide. If you are looking for mathematics help then share your requirements at info@mathassignmenteperts.com Matrices and Determinants Question : Solve for, y and z Augmented Matri: y 6 4z z 6 y 6y z or y 4z 6 y z 6 6y z Multiply row by and add with row and get your new row ; R R R 6 Multiply with row and add with row and get your new row ; R R R 0 7 Multiply row by / gives you following ; R R 0 7 Multiply by - with row and add with row and get your row R R R ; Now by backward substitution to solve for variables: z, y () 0 ( 6) 4() 6

2 y 04 Now practice: Write the system of equations and solve. Ans (, -, ) Question : Below are three row-reduced echelon forms for matrices of certain linear systems. For each matri, tell how many solutions the system has. Eplain. If there are any, find the solutions. If there are infinitely many solutions, find the general formula and two particular solutions. [ 0 0 ] a. 0 0 The system has unique solution with =, y=-, z= 0 0 [ 0 0 ] b The system does not have solution [ 0 0 0] c. 0 0 The system has many solutions, it is dependent and consistent yz 0 The general solutions are and y z z 0 z Question : Sarrus Rule. In order to compute Note: Sarrus rule is only applicable if the determinant is of order by. Eample: Use Sarrus rule to find the value of = A A 4 4 4

3 4 + + = 9 Question 4: Finding inverse of a matri: Now let s learn how to find inverse of a matri. There are different methods to find inverse matri. Method. Use Shortcut for by matri a b d b Let A c d then A ad bc c a Eample: A then A 9 9 ( ) () 9 9 Method (Optional). Use Gauss-Jordan elimination to transform [ A I ] into [ I A - ]. Eample: Consider a matri A 4 and write the following Method : Adjoint method A - = (adjoint of A) or A - = (cofactor matri of A) T Let 4 A 4, then A and A Now we know how to find inverse, let s go back to solution of system of equations:

4 Eample : (continued) given system is y y If we write in matri form then we get the following, A X y B If we do AX B, we get the given system and we can rewrite AX B X A B We have seen that A then A 9 9 ( ) () 9 9 Now 9 9 y 9 9 so and y Question : Use Cramer s Rule to solve the system: 4 - y + z = - + y + z = 0 y + 6z = Solution. We begin by setting up four determinants: D consists of the coefficients of, y, and z from the three equations :

5 is obtained by replacing the -coefficients in the first column of D with the constants from the right sides of the equations. is obtained by replacing the y-coefficients in the second column of D with the constants from the right sides of the equations. is obtained by replacing the z-coefficients in the third column of D with the constants from the right sides of the equations. Net, we evaluate the four determinants: = 4( (-6)) + ( ) + (-4 0) = 4(8) + (-) + (-4) = 7 4 = = -( (-6)) + (60 ) + (-0 ) = -(8)+(7) + (-) = = - = 4(60 ) + ( ) + ( 0) = 4(7) + (-) + (-48) = 8-48 = 6

6 = 4( (-0)) + ( 0) (-4 0) = 4() + (-48) (-4) = = 0 Substitute these four values into the formula from Cramer s Rule: So, the solution is (-,, ). Question 6: Solve the system Solution: Now applying the operation r r R we have the following 6 Applying / r R we have 6.. And by r r R.... Finally we the following by applying /. r R..

7 We now have that, and other unknowns can easily be found by backward substitution into second and first equations. We have the solution,, ) (,, ). This method is called the Gaussian Elimination method. Question 7: A B 0 C ( 6 D E 4 F G Order of above matrices: A is a matri by, B is by, C is by, D is by, E is by, F and G are by matrices. Find the following if possible. -B,. A-C,. F+G, 4. B-C Answers: B A-C F G B-C is not possible