Examine the consistency of the system of equations: 3x - y - 2z = 2…

Question 38 of 58intermediate🔧 ApplyLong Answer4 marks

Examine the consistency of the system of equations: 3x - y - 2z = 2 2y - z = -1 3x - 5y = 3

Correct Answer

The system of equations is inconsistent.

Exercise: EXERCISE 4.5 | Q: 5 | (Page 22)

For More Understanding

Explanation

To examine the consistency, we write the system in the form $AX = B$ . We calculate the determinant of the coefficient matrix $A$ . If $|A| \neq 0$ , the system is consistent with a unique solution. If $|A| = 0$ , we calculate $(adj A)B$ . If $(adj A)B \neq 0$ , the system is inconsistent (solution does not exist). Here, $|A| = 0$ and $(adj A)B \neq 0$ , leading to the conclusion that the system is inconsistent.

Solution Steps

Step 1: Write the system in matrix form $AX = B$ . $A = \begin{bmatrix} 3 & -1 & -2 \\ 0 & 2 & -1 \\ 3 & -5 & 0 \end{bmatrix}$ , $X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$ , and $B = \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$ .
Step 2: Calculate the determinant of matrix $A$ ( $|A|$ ). Expanding along Row 1:
$|A| = 3(0 - 5) - (-1)(0 - (-3)) + (-2)(0 - 6)$ $|A| = 3(-5) + 1(3) - 2(-6)$
$|A| = -15 + 3 + 12 = 0$ . Since $|A| = 0$ , the matrix $A$ is singular, and the system does not have a unique solution.
Step 3: Calculate the adjoint of $A$ ( $adj A$ ). First, find the cofactors of $A$ :
$C_{11} = -5, C_{12} = -3, C_{13} = -6$ $C_{21} = 10, C_{22} = 6, C_{23} = 12$ $C_{31} = 5, C_{32} = 3, C_{33} = 6$ $adj A = \begin{bmatrix} -5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6 \end{bmatrix}$
Step 4: Calculate the product $(adj A)B$ . $(adj A)B = \begin{bmatrix} -5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6 \end{bmatrix} \begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix} = \begin{bmatrix} -10 - 10 + 15 \\ -6 - 6 + 9 \\ -12 - 12 + 18 \end{bmatrix} = \begin{bmatrix} -5 \\ -3 \\ -6 \end{bmatrix} \neq 0$ .
Step 5: Conclusion. Since $|A| = 0$ and $(adj A)B \neq 0$ , the solution does not exist. Therefore, the system is inconsistent.

Previous Next