If [] denotes the greatest integer less than or equal to the real number under consideration and `-1lt=xlt0,0lt=ylt1,1lt=zlt2,` the value of the determinant `|{:([x]+1,[y],[z]),([x],[y]+1,[z]),([x],[y],[z]+1):}|` is

Options

A. [x]
B. [y]
C. [z]
D. None of these

Answer

To solve the determinant problem step by step, we start by analyzing the given ranges for \( x \), \( y \), and \( z \):

1. **Identify the greatest integer values**:
- Since \( -1 < x < 0 \), we have \( \lfloor x \rfloor = -1 \).
- Since \( 0 < y < 1 \), we have \( \lfloor y \rfloor = 0 \).
- Since \( 1 < z < 2 \), we have \( \lfloor z \rfloor = 1 \).

2. **Substituting the values into the determinant**:
The determinant can be expressed as:
\[ | \begin{array}{ccc} \lfloor x \rfloor + 1 & \lfloor y \rfloor & \lfloor z \rfloor \\ \lfloor x \rfloor & \lfloor y \rfloor + 1 & \lfloor z \rfloor \\ \lfloor x \rfloor & \lfloor y \rfloor & \lfloor z \rfloor + 1 \end{array} | \]
Substituting the values we found:
\[ | \begin{array}{ccc} -1 + 1 & 0 & 1 \\ -1 & 0 + 1 & 1 \\ -1 & 0 & 1 + 1 \end{array} | \]
This simplifies to:
\[ | \begin{array}{ccc} 0 & 0 & 1 \\ -1 & 1 & 1 \\ -1 & 0 & 2 \end{array} | \]

3. **Calculating the determinant**:
We can calculate the determinant using the formula for a 3x3 matrix:
\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where the matrix is:
\[ \begin{array}{ccc} a & b & c \\ d & e & f \\ g & h & i \end{array} \]
For our matrix:
\[ a = 0, b = 0, c = 1, d = -1, e = 1, f = 1, g = -1, h = 0, i = 2 \]
Plugging in the values:
\[ |A| = 0(1 \cdot 2 - 1 \cdot 0) - 0(-1 \cdot 2 - 1 \cdot -1) + 1(-1 \cdot 0 - 1 \cdot -1) \]
Simplifying this gives:
\[ = 0 - 0 + 1(0 + 1) = 1 \]

4. **Final value of the determinant**:
Thus, the value of the determinant is \( 1 \).