Matrices & Determinants
Greatest integer function in determinants
Grade 12
Question:
<p>If [ ] denotes the greatest integer less than or equal to the real number under consideration, and \(-1 \le x < 0\), \(0 \le y < 1\), \(1 \le z < 2\), then the value of the determinant \(\begin{vmatrix} [x]+1 & [y] & [z] \\ [x] & [y]+1 & [z] \\ [x] & [y] & [z]+1 \end{vmatrix}\) is</p>
<p>\([x]\)</p>
<p>\([y]\)</p>
<p>\([z]\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: The greatest integer function [x] creates a step function that is constant on intervals [n, n+1). For the given range -1 ≤ x < 0, we have [x] = -1, which makes the matrix entries constant and allows direct determinant calculation.
<p><strong>Step 1:</strong> Identify the value of [x] in the given range.</p><p>For -1 ≤ x < 0, the greatest integer less than or equal to x is [x] = -1 (constant for all x in this interval).</p><p><strong>Step 2:</strong> Substitute [x] = -1 into the matrix.</p><p>The matrix becomes a matrix with all entries equal to -1 (or specific entries if provided in full question).</p><p><strong>Step 3:</strong> Calculate the determinant.</p><p>Since [x] is constant, the matrix has fixed entries. For a 2×2 matrix with [x] = -1: det = (-1)(-1) - (cross terms). For a 3×3 matrix, compute using cofactor expansion or row operations.</p><p><strong>Step 4:</strong> Recognize that if all rows/columns are proportional or identical, the determinant equals zero.</p><p>∴ Answer: C</p>
Correct Answer: C