Question

\(\begin{vmatrix} a+b+2c & a & b \\[0.3em] c & b+c+2c & b \\[0.3em] c & a & c+a2b \end{vmatrix}\)

• A. \( (a + b + c)^3 \)
• B. \( 2(a + b + c)^3 \)
• C. \( 3(a + b + c)^3 \)
• D. \( (a + b + c) \)

Hint:

For matrix determinants, use cofactor expansion and carefully simplify the 2x2 determinants. After simplifying, check the final expression for patterns like cubes or squares in the terms.

Answer 1

The Correct Option is B

We are given the matrix: \[ A = \begin{pmatrix} a + b + 2c & a & b
c & b + c + 2a & b
c & a & c + a + 2b \end{pmatrix} \] To find the determinant of this matrix, we use cofactor expansion along the first row: \[ \text{det}(A) = (a + b + 2c) \cdot \begin{vmatrix} b + c + 2a & b
a & c + a + 2b \end{vmatrix} - a \cdot \begin{vmatrix} c & b
c & c + a + 2b \end{vmatrix} + b \cdot \begin{vmatrix} c & b + c + 2a
c & a \end{vmatrix} \] After performing the calculations for each of the 2x2 determinants and simplifying the terms, we find that: \[ \text{det}(A) = 2(a + b + c)^3 \] Thus, the value of the determinant is \( 2(a + b + c)^3 \).

Answer 2

Problem: Evaluate the determinant \[ \begin{vmatrix} a + b + 2c & a & b \\ c & b + c + 2c & b \\ c & a & c + a + 2b \end{vmatrix}. \]

Step 1: Simplify the matrix entries Note that \( b + c + 2c = b + 3c \) and \( c + a + 2b = a + 2b + c \). So the matrix becomes: \[ \begin{vmatrix} a + b + 2c & a & b \\ c & b + 3c & b \\ c & a & a + 2b + c \end{vmatrix}. \]

Step 2: Perform row operations (if needed) or calculate determinant directly By expanding or using row operations, the determinant evaluates to: \[ 2 (a + b + c)^3. \]

Final answer: \[ \boxed{2 (a + b + c)^3}. \]