Choose the incorrect statement

Options

A. A matrix A = [3 ] is a scalar matrix of order 1
B. A matrix B = `[(-1,0),( 0,-1)]` is a scalar matrix of order 2
C. A matrix `C=[(sqrt(3),0,0),(0,sqrt(3),0),(0,0,sqrt(3))]` of order 3 is not a scalar matrix
D. None of these

Answer

To solve the problem of identifying the incorrect statement regarding scalar matrices, we will follow these steps:

### Step 1: Understand the Definition of a Scalar Matrix
A scalar matrix is defined as a square matrix where all the diagonal elements are equal and all off-diagonal elements are zero. For a matrix of order n, it can be represented as:
\[ A = \begin{pmatrix} a & 0 & 0 & \ldots & 0 \\ 0 & a & 0 & \ldots & 0 \\ 0 & 0 & a & \ldots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \ldots & a \end{pmatrix} \]
where \( a \) is a scalar.

### Step 2: Analyze the Given Options
We need to evaluate the statements provided in the options to determine which one is incorrect.

1. **Option 1:** A matrix with diagonal elements all equal to -1 is a scalar matrix.
2. **Option 2:** A matrix with diagonal elements all equal to 3 is a scalar matrix.
3. **Option 3:** A matrix with diagonal elements a11, a22, a33 all equal to a is a scalar matrix.
4. **Option 4:** A matrix of order three that does not have all diagonal elements equal is not a scalar matrix.

### Step 3: Evaluate Each Statement
- **Option 1:** True. If all diagonal elements are -1 and off-diagonal elements are 0, it is a scalar matrix.
- **Option 2:** True. If all diagonal elements are 3 and off-diagonal elements are 0, it is a scalar matrix.
- **Option 3:** True. If all diagonal elements are equal to a, it is a scalar matrix.
- **Option 4:** True. If a matrix of order three has diagonal elements that are not all equal, it cannot be a scalar matrix.

### Step 4: Identify the Incorrect Statement
Since all the options provided are true regarding scalar matrices, we need to check if there is any misunderstanding in the question. However, based on the analysis, if all options are indeed true, then the question may not have an incorrect statement as stated.

### Conclusion
Upon reviewing the options, it appears that there is no incorrect statement provided, as all statements align with the definition of scalar matrices. If we have to choose one based on the context of the question, we might conclude that the misunderstanding lies in the phrasing of the question itself.