Matrices & Determinants
System of linear equations — matrix method
Grade Class 12
Question:
Let S be the set of all column matrices \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} such that b_1, b_2, b_3 \in \mathbb{R} and the system of equations (in real variables) <br> -x + 2y + 5z = b_1 <br> 2x - 4y + 3z = b_2 <br> x - 2y + 2z = b_3 <br> has at least one solution. Then, which of the following system(s) (in real variables) has (have) at least one solution of each \begin{pmatrix} b_1 \\ b_2 \\ b_3 \end{pmatrix} \in S?
(A) x + 2y + 3z = b_1, 4y + 5z = b_2 \text{ and } x + 2y + 6z = b_3
(B) x + y + 3z = b_1, 5x + 2y + 6z = b_2 \text{ and } -2x - y - 3z = b_3
(C) -x + 2y - 5z = b_1, 2x - 4y + 10z = b_2 \text{ and } x - 2y + 5z = b_3
(D) x + 2y + 5z = b_1, 2x + 3z = b_2 \text{ and } x + 4y - 5z = b_3
Step-by-Step Solution
Key Concept: The system has a solution if the augmented matrix [A|B] is consistent. For the given system, the determinant of the coefficient matrix is 0. Row operations show that the condition for consistency is b1 + b3 = b2.
The coefficient matrix A = [[-1, 2, 5], [2, -4, 3], [1, -2, 2]]. Performing row operations R2 -> R2 + 2R1 and R3 -> R3 + R1, we get [[-1, 2, 5], [0, 0, 13], [0, 0, 7]]. The system is consistent if the same operations on the augmented matrix result in a consistent system. Specifically, the condition for consistency is b2 + 2b1 = 13/7(b3 + b1), which simplifies to b1 + b3 = b2. We check each option to see if they satisfy this condition.
Correct Answer: 1, 3