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 task is to determine if the provided Assertion (A) and Reason (R) regarding matrices are correct, and if Reason (R) sufficiently explains Assertion (A). The statements are as follows:

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

Reason (R): A diagonal matrix is classified as a scalar matrix if all its non-zero diagonal elements are identical.

Let's define the terms:

• Diagonal Matrix: A matrix where all entries not on the main diagonal are zero. For matrix \( A = \text{diag} [3, 5, 2] \), its representation is:

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

• This fits the definition of a diagonal matrix.
• Scalar Matrix: A specific category of diagonal matrix where all elements on the main diagonal are equal.

In the matrix \( A = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{bmatrix} \), the diagonal elements \(3\), \(5\), and \(2\) are not equal. Consequently, \( A \) is not a scalar matrix. Therefore, Assertion (A) is false.

Now, let's evaluate Reason (R):

• Reason (R) correctly states that a diagonal matrix is a scalar matrix if all its non-zero diagonal elements are the same. This aligns with the definition of a scalar matrix.

Conclusion: Assertion (A) is false, while Reason (R) is true. The correct option is: Assertion (A) is false but Reason (R) is true.