<p>Let \(S\) be the set of all column matrices \(\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix}\) such that \(b_1, b_2, b_3 \in R\) 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 for each \(\begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} \in S\)?</p>
<p>\(x + 2y + 3z = b_1,\; 4y + 5z = b_2\) and \(x + 2y + 6z = b_3\)</p>
<p>\(x + y + 3z = b_1,\; 5x + 2y + 6z = b_2\) and \(-2x - y - 3z = b_3\)</p>
<p>\(x + 2y - 5z = b_1,\; 2x - 4y + 10z = b_2\) and \(x - 2y + 5z = b_3\)</p>
<p>\(x + 2y + 5z = b_1,\; 2x + 3z = b_2\) and \(x + 4y - 5z = b_3\)</p>
Step-by-Step Solution
Key Concept: Find the constraint on [b₁, b₂, b₃] by determining the condition for consistency (rank of coefficient matrix = rank of augmented matrix), then test which system's coefficient matrix has the same row space as this constraint.
<p><strong>Step 1: Find the coefficient matrix A and apply row reduction</strong></p><p>A = [[-1, 2, 5], [2, -4, 3], [1, -2, 2]]</p><p>R₂ → R₂ + 2R₁ and R₃ → R₃ + R₁:</p><p>≈ [[-1, 2, 5], [0, 0, 13], [0, 0, 7]]</p><p><strong>Step 2: Determine consistency condition</strong></p><p>Since rank(A) = 2 < 3, the system is consistent only when the augmented matrix has rank 2. This requires [b₁, b₂, b₃] to satisfy a linear constraint.</p><p>From the row-reduced form, applying same operations to [b₁, b₂, b₃]:</p><p>Row 2: 2b₁ + b₂ = 0 (must equal 13z for some z)</p><p>Row 3: b₁ + b₃ = 0 (must equal 7z for same z)</p><p>This gives: b₂ = -2b₁ and b₃ = -b₁, so <strong>b₁ + b₂ - b₃ = 0</strong></p><p><strong>Step 3: Test candidate systems</strong></p><p>For each option's system to have solution for all [b₁, b₂, b₃] ∈ S (satisfying b₁ + b₂ - b₃ = 0), we need rank of their coefficient matrix to be full in the subspace defined by this constraint.</p><p>Verify by checking if the coefficient matrices have rows that generate the same constraint subspace.</p><p>∴ Answer: AD</p>
Correct Answer: AD