Evaluate the determinant: left| begin{array}{ | vedclass

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(D) Let $\Delta = \left| \begin{array}{ccc} x & 4 & y + z \ y & 4 & z + x \ z & 4 & x + y \end{array} ight|$.Apply the column operation $C_1 	o C

Vedclass · Accepted Answer

$0$

Vedclass · Answer

(D) Let $\Delta = \left| \begin{array}{ccc} x & 4 & y + z \ y & 4 & z + x \ z & 4 & x + y \end{array} ight|$.Apply the column operation $C_1 	o C_1 + C_3$:$\Delta = \left| \begin{array}{ccc} x + y + z & 4 & y + z \ y + z + x & 4 & z + x \ z + x + y & 4 & x + y \end{array} ight|$.Take $(x + y + z)$ common from $C_1$:$\Delta = (x + y + z) \left| \begin{array}{ccc} 1 & 4 & y + z \ 1 & 4 & z + x \ 1 & 4 & x + y \end{array} ight|$.Since $C_1$ and $C_2$ are proportional (or specifically,$C_1$ is a multiple of the vector $(1, 1, 1)^T$ and $C_2$ is $4$ times the same vector),the two columns are linearly dependent.Alternatively,since $C_1$ and $C_2$ are proportional,the value of the determinant is $0$.

Evaluate the determinant: $\left| \begin{array}{ccc} x & 4 & y + z \\ y & 4 & z + x \\ z & 4 & x + y \end{array} \right|$

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