Question 44 of 58intermediate🔧 ApplyLong Answer4 marks

Solve system of linear equations, using matrix method: 2x + y + z = 1 x - 2y - z = 3/2 3y - 5z = 9

Correct Answer

x = 1, y = 1/2, z = -3/2

Exercise: EXERCISE 4.5 | Q: 11 | (Page 22)
For More Understanding

Explanation

This question is from Exercise 4.5, Question 11. The solution follows the matrix method demonstrated in Example 17 of the textbook, where the system AX = B is solved by finding A^(-1) and computing X = A^(-1)B. The key steps involve writing equations in matrix form, calculating determinant to verify inverse exists, finding cofactors and adjoint, then multiplying by the constant matrix.

Solution Steps

  1. Step 1: Write the system in matrix form AX = B

  2. A = [[2, 1, 1], [1, -2, -1], [0, 3, -5]], X = [[x], [y], [z]], B = [[1], [3/2], [9]]

  3. Step 2: Calculate |A|

  4. |A| = 2[(-2)(-5) - (-1)(3)] - 1[(1)(-5) - (-1)(0)] + 1[(1)(3) - (-2)(0)] = 2(13) + 5 + 3 = 34 ≠\neq 0

  5. Since |A| ≠\neq 0, A is nonsingular and inverse exists.

  6. Step 3: Find cofactors of A

  7. C₁₁ = 13, C₁₂ = 5, C₁₃ = 3

  8. C₂₁ = 8, C₂₂ = -10, C₂₃ = -6

  9. C₃₁ = 1, C₃₂ = 3, C₃₃ = -5

  10. Step 4: Find adj(A) and A^(-1)

  11. adj(A) = [[13, 8, 1], [5, -10, 3], [3, -6, -5]]

  12. A^(-1) = (1/34)[[13, 8, 1], [5, -10, 3], [3, -6, -5]]

  13. Step 5: Calculate X = A^(-1)B

  14. x = (1/34)[13(1) + 8(3/2) + 1(9)] = (1/34)(34) = 1

  15. y = (1/34)[5(1) + (-10)(3/2) + 3(9)] = (1/34)(17) = 1/2

  16. z = (1/34)[3(1) + (-6)(3/2) + (-5)(9)] = (1/34)(-51) = -3/2

  17. Hence, x = 1, y = 1/2, z = -3/2

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