Finding Determinant of Matrix A=[R1:b+c,a,bc and R2:b,c+a,b and R3:c,c,a+b] Using Properties

Math Problem Statement

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:

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? --- 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.

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Matrix Algebra
Determinants
Row Operations
Linear Dependence

Determinant of a matrix with dependent rows

Advanced High School

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