Question

Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

• A. Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true but Reason (R) is false.
• D. Assertion (A) is false but Reason (R) is true.

Hint:

A scalar matrix is a special type of diagonal matrix where all diagonal elements are equal. If they are not equal, the matrix is just a diagonal matrix.

Answer 1

The Correct Option is D

The problem requires evaluating whether given statements about matrices are true, and if the Reason (R) appropriately explains the Assertion (A). Let's dissect the given statements:

Assertion (A): \( A = \text{diag} [3, 5, 2] \) is a scalar matrix of order \( 3 \times 3 \).

Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.

First, let's clarify definitions:

• Diagonal Matrix: A matrix in which the entries outside the main diagonal are all zero. For matrix \( A = \text{diag} [3, 5, 2] \), it looks like this:

\[ A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix} \]

• This is indeed a diagonal matrix.
• Scalar Matrix: A special type of diagonal matrix where all diagonal elements are equal.

In \( A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix} \), the diagonal elements are not equal (\(3 \neq 5 \neq 2\)). Thus, \( A \) is not a scalar matrix. Hence, the assertion that \( A \) is a scalar matrix is false.

Now, let's assess the given reason:

• The Reason (R) states that if a diagonal matrix has all non-zero elements equal, it becomes a scalar matrix. This is true based on the definition of a scalar matrix.

Conclusion: The assertion is false, but the reason is true. The correct choice is: Assertion (A) is false but Reason (R) is true.