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Statement -1 The value of the determent ...

Statement -1 The value of the determent `|{:(1,2,3),(4,5,6),(7,8,0):}|ne 0`
Statement -2 Neither of two rows rows or columns of `|{:(1,2,3),(4,5,6),(7,8,0):}|` is identical.

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To solve the given problem, we need to evaluate the determinant and analyze the statements provided. Let's break it down step by step. ### Step 1: Write down the determinant We have the determinant: \[ D = \begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 0 \end{vmatrix} \] ### Step 2: Expand the determinant We can expand this determinant along the first row. The formula for the determinant of a 3x3 matrix is: \[ D = a(ei - fh) - b(di - fg) + c(dh - eg) \] where the matrix is: \[ \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} \] For our determinant: - \(a = 1\), \(b = 2\), \(c = 3\) - \(d = 4\), \(e = 5\), \(f = 6\) - \(g = 7\), \(h = 8\), \(i = 0\) Plugging in these values, we get: \[ D = 1 \cdot (5 \cdot 0 - 6 \cdot 8) - 2 \cdot (4 \cdot 0 - 6 \cdot 7) + 3 \cdot (4 \cdot 8 - 5 \cdot 7) \] ### Step 3: Calculate each term Now we calculate each term: 1. \(1 \cdot (0 - 48) = 1 \cdot (-48) = -48\) 2. \(-2 \cdot (0 - 42) = -2 \cdot (-42) = 84\) 3. \(3 \cdot (32 - 35) = 3 \cdot (-3) = -9\) ### Step 4: Combine the results Now, we combine the results: \[ D = -48 + 84 - 9 = 27 \] ### Step 5: Conclusion about Statement 1 Since \(D = 27\) which is not equal to zero, **Statement 1 is true**. ### Step 6: Analyze Statement 2 Statement 2 claims that neither of the two rows or columns of the determinant is identical. - The rows are: - Row 1: (1, 2, 3) - Row 2: (4, 5, 6) - Row 3: (7, 8, 0) - The columns are: - Column 1: (1, 4, 7) - Column 2: (2, 5, 8) - Column 3: (3, 6, 0) None of the rows or columns are identical, therefore **Statement 2 is also true**. ### Step 7: Check the explanation Finally, we need to check if Statement 2 is the correct explanation for Statement 1. While Statement 2 is true, it does not necessarily imply that the determinant is non-zero. There can be cases where the determinant is non-zero even if rows or columns are not identical. ### Final Conclusion - **Statement 1 is true.** - **Statement 2 is true.** - **Statement 2 is not the correct explanation for Statement 1.**
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