Question

If \(A=(a_{ij})\) is a \(3\times 3\) diagonal matrix such that \(a_{11}=1\), \(a_{22}=2\) and \(a_{33}=3\), then \(|A|\)

• A. \(6\)
• B. \(7\)
• C. \(-6\)
• D. \(-7\)

Hint:

For diagonal, upper triangular, or lower triangular matrices, determinant is equal to the product of diagonal entries.

Answer 1

The Correct Option is A

Concept:
A diagonal matrix is a matrix in which all non-diagonal elements are zero. For example: \[ A= \begin{bmatrix} a_{11} & 0 & 0\\ 0 & a_{22} & 0\\ 0 & 0 & a_{33} \end{bmatrix} \] The determinant of a diagonal matrix is the product of its diagonal elements. So: \[ |A|=a_{11}a_{22}a_{33} \]

Step 1: Write the diagonal elements.
Given: \[ a_{11}=1 \] \[ a_{22}=2 \] \[ a_{33}=3 \] So the matrix is: \[ A= \begin{bmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{bmatrix} \]

Step 2: Use determinant property.
For a diagonal matrix: \[ |A|=\text{product of diagonal elements} \] Therefore: \[ |A|=1\times2\times3 \]

Step 3: Calculate.
\[ |A|=6 \]

Step 4: Check the options.
Option (A) \(6\) is correct.
Option (B) \(7\) is incorrect.
Option (C) \(-6\) is incorrect because all diagonal entries are positive.
Option (D) \(-7\) is incorrect. Hence, the correct answer is: \[ \boxed{(A)\ 6} \]