What is the determinant of the matrix? | x y y+z | | z x z+x | | y z x+y |
(x - z) 2 (x + y + z)
Quest
Find the determinant of the matrix:
| x y y+z | | z x z+x | | y z x+y |
Solution (steps)
- Write the matrix and apply the column operation
C3 ← C3 - C1 - C2. This operation does not change the determinant. - Compute the new third column:C3' = (y+z) - x - y = z - x (z+x) - z - x = 0 (x+y) - y - z = x - z So C3' = [z-x, 0, x-z]^T = (z-x)[1, 0, -1]^T
- Factor out (z-x) from the third column. The determinant becomes(z - x) * det( [ x y 1 ] [ z x 0 ] [ y z -1 ] )
- Evaluate the remaining 3×3 determinant. Expanding or computing directly yields
(z - x)(x + y + z)(z - x), i.e. it contributes another factor of (z-x). - Therefore the full determinant equals
\((x - z)^2 (x + y + z)\)
Final boxed answer: (x - z)2(x + y + z)
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Select the correct answer using the code given below:
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Important Questions from Determinants
The value of determinant \(\left| {\begin{array}{*{20}{c}} {a - b - c}&{2a}&{2a}\\ {2b}&{b - c - a}&{2b}\\ {2c}&{2c}&{c - a - b} \end{array}} \right|\) is:
If \(\left| {\begin{array}{*{20}{c}} 5&a\\ a&2 \end{array}} \right| = \left| {\begin{array}{*{20}{c}} 2&1\\ 3&2 \end{array}} \right|\), then the values of a are:
The value of the determinant \(\left| {\begin{array}{*{20}{c}} {\sqrt {13} + \sqrt 3 }&{2\sqrt 5 }&{\sqrt 5 }\\ {\sqrt {15} + \sqrt {26} }&5&{\sqrt {10} }\\ {3 + \sqrt {65} }&{\sqrt {15} }&5 \end{array}} \right|\) is
The determinant \(\left| {\begin{array}{*{20}{c}} {xp + y}&x&y\\ {yp + z}&y&z\\ 0&{xp + y}&{yp + z} \end{array}} \right| = 0,\) if
If Δ(a, b, c, α) = 0 for every α > 0, then which one of the following is correct ?
