Find the determinent using properties if the matrix A=[R1:b+c,a,bc and R2:b,c+a,b and R3:c,c,a+b]

To find the determinant of the given \(3 \times 3\) matrix \(A\) using properties, let's first express the matrix explicitly:

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

### Step 1: Simplify using row operations
We can use row operations that do not change the determinant value. Let's subtract the second row from the first row:

\[
R1' = R1 - R2 = \begin{pmatrix} b+c & a & bc \end{pmatrix} - \begin{pmatrix} b & c+a & b \end{pmatrix} = \begin{pmatrix} c & -a & bc-b \end{pmatrix}
\]

Now, the matrix becomes:

\[
A' = \begin{pmatrix}
c & -a & bc-b \\
b & c+a & b \\
c & c & a+b
\end{pmatrix}
\]

### Step 2: Apply properties of determinants
Next, observe that the third row \( R3 \) can be written as a linear combination of the other two rows:

\[
R3 = R1 + R2
\]

This means the third row is dependent on the other rows, implying that the determinant of the matrix is 0.

### Conclusion
Since the determinant of a matrix with dependent rows is zero, the determinant of the matrix \( A \) is:

\[
\text{det}(A) = 0
\]

Would you like a deeper explanation or have any questions about this solution?

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Here are some related questions:
1. How can we determine the rank of a matrix with dependent rows?
2. What happens to the determinant if we swap two rows of a matrix?
3. How do row operations affect the determinant of a matrix?
4. Can a matrix with a zero determinant be inverted?
5. What is the geometric interpretation of a matrix with a determinant of zero?

**Tip:** When dealing with determinants, always check for row or column dependencies to simplify your calculations.

Learn how to find the determinant of a 3x3 matrix A=[R1:b+c,a,bc and R2:b,c+a,b and R3:c,c,a+b] using row operations and properties of determinants. Understand why the determinant is 0 when rows are linearly dependent. Suitable for advanced high school students and beyond.

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