Choose the correct answer

Options

A. Every identity matrix is scalar matrix
B. Every scalar matrix is and identity matrix
C. Every diagonal matrix is an identity matrix
D. A square matrix whose each element is 1 is an identity matrix

Answer

To solve the question, we need to analyze each statement regarding matrices and determine their validity. Let's break down the statements one by one.

### Step-by-Step Solution:

1. **Understanding Identity Matrix**:
- An identity matrix is a square matrix where all the diagonal elements are 1, and all non-diagonal elements are 0. For example, a 3x3 identity matrix looks like this:
\[ I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

**Hint**: Recall that the identity matrix acts as the multiplicative identity in matrix multiplication.

2. **Understanding Scalar Matrix**:
- A scalar matrix is a square matrix where all the diagonal elements are equal (let's say k), and all non-diagonal elements are 0. For example, a 3x3 scalar matrix could look like this:
\[ S = \begin{pmatrix} k & 0 & 0 \\ 0 & k & 0 \\ 0 & 0 & k \end{pmatrix} \]

**Hint**: Remember that a scalar matrix is a special case of a diagonal matrix where all diagonal elements are equal.

3. **Understanding Diagonal Matrix**:
- A diagonal matrix is a matrix where all non-diagonal elements are 0, but the diagonal elements can be any values (not necessarily equal). For example:
\[ D = \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & c \end{pmatrix} \]

**Hint**: Diagonal matrices can have different values on the diagonal.

4. **Analyzing Each Statement**:
- **Statement A**: "Every identity matrix is a scalar matrix."
- This is **True** because an identity matrix has all diagonal elements equal to 1, which satisfies the definition of a scalar matrix.

- **Statement B**: "Every scalar matrix is an identity matrix."
- This is **False** because a scalar matrix can have diagonal elements equal to any constant k, not just 1.

- **Statement C**: "Every diagonal matrix is an identity matrix."
- This is **False** because diagonal matrices can have different values on the diagonal.

- **Statement D**: "A square matrix whose each element is one is an identity matrix."
- This is **False** because a matrix with all elements equal to 1 does not have 0s in the non-diagonal positions, thus it is not an identity matrix.

5. **Conclusion**:
- The only true statement is **Statement A**: "Every identity matrix is a scalar matrix."

### Final Answer:
The correct answer is **Statement A**.