If `A=[a_(ij)]` is a scalar matrix, then trace of A is

Options

A. `undersetisumundersetjsuma_(ij)`
B. `underset isuma_(ij)`
C. `undersetjsuma_(ij)`
D. `undersetisuma_(ij)`

Answer

To find the trace of a scalar matrix \( A = [a_{ij}] \), we can follow these steps:

### Step 1: Understand the definition of a scalar matrix
A scalar matrix is a special type of diagonal matrix where all the diagonal elements are equal. Thus, we can express a scalar matrix \( A \) as:
\[ A = \lambda I \]
where \( \lambda \) is a scalar and \( I \) is the identity matrix.

### Step 2: Write the identity matrix
For a 3x3 identity matrix, it is represented as:
\[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \]

### Step 3: Formulate the scalar matrix
Since \( A \) is a scalar matrix, we can express it as:
\[ A = \lambda \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} \lambda & 0 & 0 \\ 0 & \lambda & 0 \\ 0 & 0 & \lambda \end{bmatrix} \]

### Step 4: Calculate the trace of the matrix
The trace of a matrix is defined as the sum of its diagonal elements. Therefore, for the scalar matrix \( A \):
\[ \text{Trace}(A) = a_{11} + a_{22} + a_{33} = \lambda + \lambda + \lambda = 3\lambda \]

### Conclusion
Thus, the trace of the scalar matrix \( A \) is:
\[ \text{Trace}(A) = 3\lambda \]