What type of a matrix is `A=[(0,0,5),(0,5,0),(5,0,0)]`?

Options

A. Identity matrix
B. Upper triangular matrix
C. Square matrix
D. Lower triangular matrix

Answer

To determine the type of the matrix \( A = \begin{pmatrix} 0 & 0 & 5 \\ 0 & 5 & 0 \\ 5 & 0 & 0 \end{pmatrix} \), we will analyze its properties step by step.

### Step 1: Identify the dimensions of the matrix
The matrix \( A \) has 3 rows and 3 columns, which means it is a square matrix.

**Hint:** A matrix is square if the number of rows is equal to the number of columns.

### Step 2: Check for the identity matrix
An identity matrix is a square matrix where all the diagonal elements are 1 and all other elements are 0. In matrix \( A \), the diagonal elements are 0, 5, and 0, which do not satisfy the identity matrix condition.

**Hint:** An identity matrix looks like \( I_n = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \).

### Step 3: Check for upper triangular matrix
An upper triangular matrix has all elements below the main diagonal equal to 0. In matrix \( A \), the elements below the main diagonal (the first column) are 0, which satisfies the condition for being upper triangular.

**Hint:** For an upper triangular matrix, elements \( a_{ij} = 0 \) for all \( i > j \).

### Step 4: Check for lower triangular matrix
A lower triangular matrix has all elements above the main diagonal equal to 0. In matrix \( A \), the elements above the main diagonal (the third row) are not all 0 (specifically, \( 5 \) is present), so it does not satisfy the condition for being lower triangular.

**Hint:** For a lower triangular matrix, elements \( a_{ij} = 0 \) for all \( i < j \).

### Conclusion
Based on the analysis, matrix \( A \) is a **square matrix** and also an **upper triangular matrix**. However, it is not an identity matrix or a lower triangular matrix.

**Final Answer:** Matrix \( A \) is an upper triangular matrix and a square matrix.