The matrix `[(0,0,5),(0,5,0),(5,0,0)]` is a scalar matrix.
State true of false. If false then what type of matrix is this ?

Answer

To determine whether the given matrix \(\begin{pmatrix} 0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0 \end{pmatrix}\) is a scalar matrix, we need to follow these steps:

### Step 1: Understand the Definition of a Scalar Matrix
A scalar matrix is defined as a square matrix in which all the elements of the principal diagonal are equal and are the same constant (scalar), while all off-diagonal elements are zero.

### Step 2: Identify the Type of Matrix
The given matrix is:
\[ \begin{pmatrix} 0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0 \end{pmatrix} \]
We can see that it is a \(3 \times 3\) matrix, which means it is a square matrix.

### Step 3: Check the Principal Diagonal Elements
The principal diagonal elements of the matrix are:
- First element (top left): \(0\)
- Second element (middle): \(5\)
- Third element (bottom right): \(0\)

So, the principal diagonal elements are \(0\), \(5\), and \(0\).

### Step 4: Analyze the Principal Diagonal Elements
For the matrix to be a scalar matrix, all principal diagonal elements must be equal and a constant. Here, the principal diagonal elements are \(0\), \(5\), and \(0\), which are not equal.

### Step 5: Conclusion
Since the principal diagonal elements are not equal, we conclude that the given matrix is **not** a scalar matrix.

### Final Answer
The statement that the matrix is a scalar matrix is **false**. The given matrix is a **square matrix**.

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