Let A be a square matrix of order 3 whose all entries are 1 and let `I_(3)` be the indentiy matrix of order 3. then the matrix A - `3I_(3)` is

Options

A. invertible
B. orthogonal
C. non-invertible
D. real skew symmetric matrix

Answer

To solve the problem, we need to find the matrix \( A - 3I_3 \) where \( A \) is a square matrix of order 3 with all entries equal to 1, and \( I_3 \) is the identity matrix of order 3.

### Step 1: Define the matrices

1. **Matrix \( A \)**:
\[ A = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} \]

2. **Identity Matrix \( I_3 \)**:
\[ I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

### Step 2: Calculate \( 3I_3 \)

Now, we multiply the identity matrix \( I_3 \) by 3:
\[ 3I_3 = 3 \times \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]

### Step 3: Calculate \( A - 3I_3 \)

Now, we subtract \( 3I_3 \) from \( A \):
\[ A - 3I_3 = \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix} - \begin{pmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{pmatrix} = \begin{pmatrix} 1-3 & 1-0 & 1-0 \\ 1-0 & 1-3 & 1-0 \\ 1-0 & 1-0 & 1-3 \end{pmatrix} \]

Calculating the entries:
\[ A - 3I_3 = \begin{pmatrix} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{pmatrix} \]

### Step 4: Determine the properties of \( A - 3I_3 \)

To check if the matrix \( A - 3I_3 \) is invertible, we need to calculate its determinant.

### Step 5: Calculate the determinant of \( A - 3I_3 \)

Using the determinant formula for a \( 3 \times 3 \) matrix:
\[ \text{det}(B) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( B = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).

For our matrix \( A - 3I_3 \):
\[ B = \begin{pmatrix} -2 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 1 & -2 \end{pmatrix} \]

Calculating the determinant:
\[ \text{det}(A - 3I_3) = -2((-2)(-2) - (1)(1)) - 1(1(-2) - (1)(1)) + 1(1(1) - (-2)(1)) \]
\[ = -2(4 - 1) - 1(-2 - 1) + 1(1 + 2) \]
\[ = -2(3) + 3 + 3 \]
\[ = -6 + 3 + 3 = 0 \]

### Conclusion

Since the determinant of \( A - 3I_3 \) is 0, the matrix \( A - 3I_3 \) is **non-invertible**.

### Final Answer
The matrix \( A - 3I_3 \) is non-invertible.