<p>Let us consider the following matrix: \[\begin{bmatrix} 1 & X & X \\ X & 1 & X \\ X & X & 1 \end{bmatrix}\] which are six non-singular matrices because six blanks (i.e., \(X\)) shall be filled by five zeros and one 1. In the same manner, we have the matrix: \[\begin{bmatrix} X & X & 1 \\ X & 1 & X \\ 1 & X & X \end{bmatrix}\] which are six non-singular matrices. Therefore, in the required case, there are more than 7. How many such matrices are possible?</p>
Step-by-Step Solution
Key Concept: Count non-singular matrices by recognizing that each matrix has exactly 6 blanks (X's) to fill with five 0's and one 1. A matrix is non-singular iff its determinant ≠ 0. The key is to systematically identify which position of the single 1 among the five 0's yields det ≠ 0 for the given patterns.
<p><strong>Step 1: Analyze Pattern 1</strong></p><p>Matrix: $\begin{bmatrix} 1 & X & X \\ X & 1 & X \\ X & X & 1 \end{bmatrix}$ with 5 zeros and 1 one among the X's.</p><p>The 6 X positions are: (1,2), (1,3), (2,1), (2,3), (3,1), (3,2).</p><p>For each position, place the single 1 and compute det. By calculation, exactly 6 configurations yield non-singular matrices (det ≠ 0).</p><p><strong>Step 2: Analyze Pattern 2</strong></p><p>Matrix: $\begin{bmatrix} X & X & 1 \\ X & 1 & X \\ 1 & X & X \end{bmatrix}$ with 5 zeros and 1 one among the X's.</p><p>The 6 X positions are: (1,1), (1,2), (2,1), (2,3), (3,2), (3,3).</p><p>Similarly computing det for each configuration, exactly 6 configurations yield non-singular matrices.</p><p><strong>Step 3: Check for Overlap</strong></p><p>The two patterns have fixed 1's at different positions (diagonal vs. antidiagonal structure), so the matrix structures are distinct. No overlap occurs between the two families.</p><p><strong>Step 4: Total Count</strong></p><p>Total non-singular matrices = 6 + 6 = <strong>12</strong></p><p>∴ Answer: C (12 such matrices are possible)</p>
Correct Answer: C