Find the Determinant [[1,a,b+c],[1,b,a+c],[1,c,a+b]]

Problem

det(1)

Solution

1. Identify the matrix and the goal to find its determinant.

det(1)

2. Apply a column operation to simplify the matrix. Add the second column to the third column ((C_3)→(C_3)+(C_2).

det(1)

3. Factor out the common term (a+b+c) from the third column.

(a+b+c)*det(1)

4. Observe that the first and third columns are identical.

(Column_1)=[[1],[1],[1]]

(Column_3)=[[1],[1],[1]]

5. Conclude that if a matrix has two identical columns, its determinant is zero.

(a+b+c)⋅0=0

Final Answer

det(1)=0

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