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.
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.
Reason (R): If a diagonal matrix has all non-zero elements equal, it is known as a scalar matrix.
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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.
Updated On: Mar 8, 2025
- Both Assertion (A) and Reason (R) are true and the Reason (R) is the correct explanation of the Assertion (A).
- Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
- Assertion (A) is true but Reason (R) is false.
- Assertion (A) is false but Reason (R) is true.
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The Correct Option is D
Solution and Explanation
Step 1: Understanding scalar matrices.
A scalar matrix is a diagonal matrix where all diagonal elements are equal, meaning: \[ A = cI \] where \( c \) is a scalar and \( I \) is the identity matrix. Step 2: Evaluating Assertion (A).
The given matrix is: \[ A = \begin{bmatrix} 3 & 0 & 0
0 & 5 & 0
0 & 0 & 2 \end{bmatrix} \] Since the diagonal elements are not equal (\( 3, 5, 2 \) are different), this is a diagonal matrix but not a scalar matrix. Thus, Assertion (A) is false. Step 3: Evaluating Reason (R).
The definition given in Reason (R) is correct. If all diagonal elements were the same, the matrix would be a scalar matrix. Hence, Reason (R) is true.
Since Assertion (A) is false and Reason (R) is true, the correct answer is (D).
A scalar matrix is a diagonal matrix where all diagonal elements are equal, meaning: \[ A = cI \] where \( c \) is a scalar and \( I \) is the identity matrix. Step 2: Evaluating Assertion (A).
The given matrix is: \[ A = \begin{bmatrix} 3 & 0 & 0
0 & 5 & 0
0 & 0 & 2 \end{bmatrix} \] Since the diagonal elements are not equal (\( 3, 5, 2 \) are different), this is a diagonal matrix but not a scalar matrix. Thus, Assertion (A) is false. Step 3: Evaluating Reason (R).
The definition given in Reason (R) is correct. If all diagonal elements were the same, the matrix would be a scalar matrix. Hence, Reason (R) is true.
Since Assertion (A) is false and Reason (R) is true, the correct answer is (D).
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