Question

The following system of equations \[ x + y + z = 1 \] \[ 2x + 3y - mz = 2 \] \[ 3x + 5y + 3z = 3 \] has no unique solution. Then the value of \( m \) is equal to

• A. \(3 \)
• B. \(5 \)
• C. \(2 \)
• D. \(-2 \)
• E. \(-3 \)

Hint:

For system of equations, no unique solution \(\Rightarrow\) determinant of coefficient matrix is zero.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A system of linear equations of the form AX = B has a "non-unique solution" (which means it has either no solution or infinitely many solutions) if and only if the determinant of the coefficient matrix A is equal to zero.
Step 2: Key Formula or Approach:
We first write the coefficient matrix A for the given system of equations. Then, we calculate its determinant and set it equal to zero to find the value of m.
The coefficient matrix A is: \[ A = \begin{pmatrix} 1 & 1 & 1
2 & 3 & -m
3 & 5 & 3 \end{pmatrix} \] We need to solve the equation det(A) = 0.
Step 3: Detailed Explanation:
We calculate the determinant of A by expanding along the first row: \[ \det(A) = 1 \begin{vmatrix} 3 & -m
5 & 3 \end{vmatrix} - 1 \begin{vmatrix} 2 & -m
3 & 3 \end{vmatrix} + 1 \begin{vmatrix} 2 & 3
3 & 5 \end{vmatrix} \] \[ \det(A) = 1((3)(3) - (-m)(5)) - 1((2)(3) - (-m)(3)) + 1((2)(5) - (3)(3)) \] \[ \det(A) = (9 + 5m) - (6 + 3m) + (10 - 9) \] \[ \det(A) = 9 + 5m - 6 - 3m + 1 \] Combine the terms: \[ \det(A) = (5m - 3m) + (9 - 6 + 1) \] \[ \det(A) = 2m + 4 \] For the system to have no unique solution, we set the determinant to zero: \[ 2m + 4 = 0 \] \[ 2m = -4 \] \[ m = -2 \] Step 4: Final Answer:
The value of m for which the system has no unique solution is -2.