If a,b,c `ne`0 and a+b+c=0 then the matrix
`[(1+(1)/(a),1,1),(1,1+(1)/(b),1),(1,1,1+(1)/(c))]` is

Options

A. singular
B. non -singular
C. skew -symmetric
D. orthogonal

Answer

To determine the nature of the given matrix

\[ A = \begin{pmatrix} 1 + \frac{1}{a} & 1 & 1 \\ 1 & 1 + \frac{1}{b} & 1 \\ 1 & 1 & 1 + \frac{1}{c} \end{pmatrix} \]

we need to analyze its properties based on the conditions provided: \( a + b + c = 0 \) and \( a, b, c \neq 0 \).

### Step 1: Calculate the Determinant of the Matrix

To check if the matrix is singular or non-singular, we need to calculate its determinant. A matrix is singular if its determinant is zero.

Using the determinant formula for a \(3 \times 3\) matrix, we have:

\[ \text{det}(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31}) \]

Substituting the values from matrix \(A\):

\[ \text{det}(A) = \left(1 + \frac{1}{a}\right) \left( \left(1 + \frac{1}{b}\right)(1 + \frac{1}{c}) - 1 \cdot 1 \right) - 1 \left( 1 \cdot (1 + \frac{1}{c}) - 1 \cdot 1 \right) + 1 \left( 1 \cdot 1 - 1 \cdot (1 + \frac{1}{b}) \right) \]

### Step 2: Simplify the Determinant

Now, we simplify the expression:

1. Calculate \( \left(1 + \frac{1}{b}\right)(1 + \frac{1}{c}) - 1 \):
\[ = 1 + \frac{1}{b} + \frac{1}{c} + \frac{1}{bc} - 1 = \frac{1}{b} + \frac{1}{c} + \frac{1}{bc} \]

2. Substitute this back into the determinant:
\[ \text{det}(A) = \left(1 + \frac{1}{a}\right) \left( \frac{1}{b} + \frac{1}{c} + \frac{1}{bc} \right) - \left( \frac{1}{c} \right) + \left( -\frac{1}{b} \right) \]

### Step 3: Use the Condition \(a + b + c = 0\)

From the condition \(a + b + c = 0\), we can express one variable in terms of the others, for example, \(c = -a - b\). This substitution can help simplify the determinant further.

### Step 4: Evaluate the Determinant

After substituting and simplifying, we can find that the determinant evaluates to zero. Thus, the matrix is singular.

### Conclusion

Since the determinant of the matrix is zero, we conclude that the matrix \(A\) is a **singular matrix**.