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The image you provided shows a matrix equation where the determinant of a matrix on the left-hand side (LHS) is equal to the product on the right-hand side (RHS). Let's break down the problem step by step.

### Problem Setup:
We have a 3x3 matrix:

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

The determinant of this matrix is said to equal:

\[
abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)
\]

### Objective:
We need to compute the determinant of the LHS matrix and check if it matches the given form on the RHS.

### Step 1: Expand the Determinant
We can use row or column operations to simplify the matrix before calculating the determinant. From the image, we see an operation: \( R_1 \rightarrow R_1 + R_2 + R_3 \).

Performing this operation on the matrix:

\[
R_1 \rightarrow (1 + a) + 1 + 1, \quad 1 + (1 + b) + 1, \quad 1 + 1 + (1 + c)
\]

This simplifies to:

\[
R_1 = (2 + a), (3 + b), (3 + c)
\]

So, the transformed matrix becomes:

\[
\begin{pmatrix}
2 + a & 3 + b & 3 + c \\
1 & 1 + b & 1 \\
1 & 1 & 1 + c
\end{pmatrix}
\]

Now, we can expand the determinant of this matrix.

### Step 2: Compute the Determinant
For a 3x3 matrix of the form:

\[
\begin{pmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{pmatrix}
\]

The determinant is given by:

\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Substitute the entries from the new matrix into this formula and simplify to match the RHS form.

### Step 3: Compare to RHS
The RHS is \( abc \left( 1 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right) \), which suggests that after expanding the determinant and simplifying, we should find this expression.

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Would you like further help in simplifying the determinant or expanding each term? Let me know if you'd like more steps or clarifications!

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### Related Questions:
1. What are the properties of determinants that allow row operations without changing the result?
2. How does cofactor expansion work for 3x3 matrices?
3. Can matrix operations like row additions affect the determinant?
4. How can we derive the form of the RHS using the properties of determinants?
5. What is the geometric interpretation of this determinant?

### Tip:
When solving for determinants of 3x3 matrices, try using row operations to simplify them as much as possible before expanding the determinant.

Solve the determinant of the 3x3 matrix and verify if it equals the given expression abc(1 + 1/a + 1/b + 1/c).

Learn how to solve the determinant of a 3x3 matrix using row operations and verify if it equals abc(1 + 1/a + 1/b + 1/c). This example covers important linear algebra concepts and determinant properties, suitable for undergraduate students.

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