Question 39 of 58intermediate🔧 ApplyLong Answer4 marks

Examine the consistency of the system of equations:

5x−y+4z=55x - y + 4z = 5 2x+3y+5z=22x + 3y + 5z = 2 5x−2y+6z=−15x - 2y + 6z = -1
Correct Answer

The system of equations is consistent and has a unique solution.

Exercise: EXERCISE 4.5 | Q: 6 | (Page 22)
For More Understanding

Explanation

To examine consistency, we write the system in matrix form AX=BAX = B and find the determinant of the coefficient matrix AA.

If ∣A∣≠0|A| \neq 0, the matrix is nonsingular, its inverse exists, and the system has a unique solution (consistent). If ∣A∣=0|A| = 0, we need to check further.

This question tests understanding of the relationship between determinant and consistency.

Solution Steps

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

    A=[5−142355−26],X=[xyz],B=[52−1]A = \begin{bmatrix} 5 & -1 & 4 \\ 2 & 3 & 5 \\ 5 & -2 & 6 \end{bmatrix}, \quad X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, \quad B = \begin{bmatrix} 5 \\ 2 \\ -1 \end{bmatrix}

  2. Step 2: Find the determinant ∣A∣|A| by expanding along the first row.

  3. Step 3:

    ∣A∣=5[3(6)−5(−2)]−(−1)[2(6)−5(5)]+4[2(−2)−3(5)]|A| = 5\big[3(6) - 5(-2)\big] - (-1)\big[2(6) - 5(5)\big] + 4\big[2(-2) - 3(5)\big]

  4. Step 4:

    ∣A∣=5(18+10)+1(12−25)+4(−4−15)|A| = 5(18 + 10) + 1(12 - 25) + 4(-4 - 15)

  5. Step 5:

    ∣A∣=5(28)+(−13)+4(−19)=140−13−76=51|A| = 5(28) + (-13) + 4(-19) = 140 - 13 - 76 = \mathbf{51}

  6. Step 6: Since ∣A∣=51≠0|A| = 51 \neq 0, matrix AA is nonsingular and its inverse exists.

  7. Step 7: Therefore, the system has a unique solution and is consistent.

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