Question 36 of 58intermediate🔧 ApplyLong Answer4 marks

Examine the consistency of the system of equations: x + 3y = 5 2x + 6y = 8

Correct Answer

The system of equations can be written in the form AX = B, where A = [1326]\begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix}, X = [xy]\begin{bmatrix} x \\ y \end{bmatrix}, and B = [58]\begin{bmatrix} 5 \\ 8 \end{bmatrix}.

Calculating the determinant of matrix A: |A| = 1(6) - 3(2) = 6 - 6 = 0. Since |A| = 0, the matrix is singular. To examine consistency, we calculate (adj A) B.

The adjoint of A is found by interchanging diagonal elements and changing signs of off-diagonal elements: adj A = [6−3−21]\begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}. Multiplying this by B: (adj A) B = [6−3−21][58]=[30−24−10+8]=[6−2]\begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} 5 \\ 8 \end{bmatrix} = \begin{bmatrix} 30 - 24 \\ -10 + 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}.

Since (adj A) B is a non-zero matrix (≠\neq O), the system has no solution. Therefore, the system is inconsistent.

Exercise: EXERCISE 4.5 | Q: 3 | (Page 22)
For More Understanding

Explanation

The question asks to examine the consistency of the system. Following the matrix method shown in Example 16 of the context, we first write the system as AX = B. We calculate the determinant |A|. Since |A| = 0, the matrix is singular. The context provides the rule: 'If (adj A) B = O, then system may be either consistent or inconsistent...'. Conversely, if (adj A) B is not equal to O, the system is inconsistent. We calculate (adj A) B and find it equals [6−2]\begin{bmatrix} 6 \\ -2 \end{bmatrix}, which is not equal to the zero matrix. Thus, the system is inconsistent.

Solution Steps

  1. Step 1: Write the system in matrix form AX = B, where A = [1326]\begin{bmatrix} 1 & 3 \\ 2 & 6 \end{bmatrix} and B = [58]\begin{bmatrix} 5 \\ 8 \end{bmatrix}.

  2. Step 2: Calculate determinant |A| = 1(6) - 3(2) = 0.

  3. Step 3: Find adj A = [6−3−21]\begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix}.

  4. Step 4: Calculate (adj A) B = [6−3−21][58]=[6−2]\begin{bmatrix} 6 & -3 \\ -2 & 1 \end{bmatrix} \begin{bmatrix} 5 \\ 8 \end{bmatrix} = \begin{bmatrix} 6 \\ -2 \end{bmatrix}.

  5. Step 5: Conclude that since (adj A) B ≠\neq O, the system is inconsistent.

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