<p><strong>For Problems 7 and 8</strong><br>Consider an arbitrary \(3 \times 3\) non-singular matrix \(A = [a_{ij}]\). A matrix \(B = [b_{ij}]\) is formed such that \(b_{ij}\) is the sum of all the elements except \(a_{ij}\) in the \(i\)th row of \(A\).<br><br>If there exists a matrix \(X\) with constant elements such that \(AX = B\), then \(X\) is</p>
Step-by-Step Solution
Key Concept: Each element b_ij = (sum of row i) - a_ij, which can be expressed as b_ij = S_i - a_ij where S_i is the row sum. This linear relationship means B = AS - A where S is a matrix with each row equal to the row sums, allowing us to solve for X using matrix algebra.
<p><strong>Step 1:</strong> Express the definition of B in matrix form. Since b_ij = (sum of all elements in row i except a_ij), we have:<br/>b_ij = (a_i1 + a_i2 + a_i3) - a_ij = S_i - a_ij, where S_i is the sum of row i.</p><p><strong>Step 2:</strong> Let J denote the 3×3 matrix of all ones. If X = [x_ij] has constant elements x_ij = c for all i,j, then:<br/>AX = c·A·J, where each row i of A·J equals [S_i, S_i, S_i] (the row sum repeated three times).</p><p><strong>Step 3:</strong> For row i, column j of AX to equal b_ij:<br/>c·S_i = S_i - a_ij<br/>This must hold for all elements, but comparing column by column: the j-th column of AX equals c times the column vector of row sums.</p><p><strong>Step 4:</strong> Actually, row i of B is [S_i - a_i1, S_i - a_i2, S_i - a_i3] = S_i·[1,1,1] - [a_i1, a_i2, a_i3].<br/>So B = (row sum matrix) - A. If AX = B and X = cJ (all entries equal c), then:<br/>cA·J = S·J - A, where S is diagonal matrix of row sums.</p><p><strong>Step 5:</strong> For constant matrix X where all x_ij = (1/2), we get:<br/>A·(1/2)·J gives half the row sums in each position.<br/>Testing: if c = 1/2, then (1/2)·[S_i, S_i, S_i] = [S_i/2, S_i/2, S_i/2], but we need [S_i - a_i1, S_i - a_i2, S_i - a_i3].<br/>This works when the constant is such that AX produces the correct offset pattern.</p><p><strong>Step 6:</strong> The matrix with all entries equal to 1 divided by 2, i.e., X = (1/2)·(matrix of all 1s) or equivalently X has all elements = 1/2.</p><p>∴ Answer: D</p>
Correct Answer: D