Saturday, July 25, 2020

Matrix Algebra - Practice Problems and Answers (#4 Part 2)

Problems:

#1 Add the matrix with row 1 given by 8, 4, 6 and row 2 given by 2, 7, 5 to the matrix with row 1 given by 2, 9, -3 and row 2 given by -4, 2/3, 8. #2 The matrix with row 1 given by 2, -9 and row 2 given by 6, 10 minus the matrix with row 1 given by 7, 4 and row 2 given by 8, -5. #3 The matrix with row 1 given by 4, 10 and row 2 given by 8, 2 times the matrix with row 1 given by 1/7, 5 and row 2 given by -3, 1. #4 (1/2) times the matrix with row 1 given by 8, -6 and row 2 given by 3, 4/7. #5 What is the inverse of the matrix with row 1 given by 8, 6 and row 2 given by 4, 9 if it can be determined? #6 What is the inverse of the matrix with row 1 given by 2, 4 and row 2 given by 4, 8 if it can be determined? #7 Solve the system of equations using matrices: 8x + 3y = 7, 9x + 2y = 4.

Answers:

#1 The matrix with row 1 given by 10, 13, 3 and row 2 given by -2, 23/3, 13. #2 The matrix with row 1 given by -5, -13 and row 2 given by -2, 15. #3 The matrix with row 1 given by -206/7, 30 and row 2 given by -34/7, 42. #4 The matrix with row 1 given by 4, -3 and row 2 given by 3/2, 2/7.

Since (8)(9) - (6)(4) = 48 which is not equal to 0, we can determine the inverse. Note that the inverse of the matrix with first row given as 8, 6 and second row given as 4, 9 is equal to 1/(48) times the matrix with first row given by 9, -6 and second row given by -4, 8 which is equal to the matrix with first row given by 3/16, -1/8 and second row given by -1/12, 1/6. #6 Since (2)(8) - (4)(4) = 0, we cannot determine the inverse. #7: We can rewrite this system as matrix with first row 8, 3 and second row 9, 2 multiplied by the matrix with first row x and second row y = the matrix with first row 7 and second row 4. Since (8)(2) - (3)(9) = -11 which is not equal to 0, the inverse exists.

Next, we need to find the inverse of matrix with row 1 given by 8, 3 and row 2 given by 9, 2. Note that the inverse of this matrix is equal to 1/((8)(2) -(3)(9)) times the matrix with row 1 given by 2, -3 and row 2 given by -9, 8 which equals the matrix with row 1 given by -2/11, 3/11 and row 2 given by 9/11 and -8/11. Thus, the matrix with row 1 being x and row 2 being y is equal to the matrix with row 1 given by -2/11, 3/11 and row 2 given by 9/11, -8/11 times the matrix with row 1 given by 7 and row 2 given by 4 which equals the matrix given by -2/11 in row 1 and 31/11 in row 2. Hence, x = -2/11 and y = 31/11.  


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