Consider the following statements:
(A) Every scalar matrix is a diagonal matrix.
(B) Every diagonal matrix is a scalar matrix.
(C) The identity matrix of order 3 is a scalar matrix.
(D) A column matrix having 3 elements is of order \(1 \times 3\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A) and (C) only
A square matrix is a diagonal matrix if all its non-diagonal elements are zero. A diagonal matrix is a scalar matrix if, in addition, all its diagonal elements are equal. A column matrix has only one column.
So a scalar matrix is a special diagonal matrix, the identity matrix has all diagonal elements equal to 1, and a column matrix with 3 elements has 3 rows and 1 column.
Checking the statements
(A) Every scalar matrix is a diagonal matrix. — Correct. Its non-diagonal elements are all zero.
(B) Every diagonal matrix is a scalar matrix. — Incorrect. \(\begin{bmatrix}1&0\\0&2\end{bmatrix}\) is diagonal but its diagonal elements are not equal.
(C) The identity matrix of order 3 is a scalar matrix. — Correct. It is diagonal with every diagonal element equal to 1.
(D) A column matrix having 3 elements is of order \(1 \times 3\). — Incorrect. It has 3 rows and 1 column, so its order is \(3 \times 1\).