Question:
\(\begin{vmatrix} a+b+2c & a & b \\[0.3em] c & b+c+2c & b \\[0.3em] c & a & c+a2b \end{vmatrix}\)
\(\begin{vmatrix} a+b+2c & a & b \\[0.3em] c & b+c+2c & b \\[0.3em] c & a & c+a2b \end{vmatrix}\)
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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.
Updated On: Mar 24, 2025
- \( (a + b + c)^3 \)
- \( 2(a + b + c)^3 \)
- \( 3(a + b + c)^3 \)
- \( (a + b + c) \)
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The Correct Option is B
Solution and Explanation
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 \( 2(a + b + c)^3 \).
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 \( 2(a + b + c)^3 \).
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