Solve please urgently needed

1.	Solve please urgently needed
Answer» $\large\underline{\sf{Solution-}}$ GIVEN determinant is TO PROVE $\rm \: \begin{gathered}\sf \left \| \begin{array}{ccc}y + z&<klux>X</klux> + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right \| \end{gathered} = {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$ Consider LHS $\rm \: = \: \: \: \begin{gathered}\sf \left \| \begin{array}{ccc}y + z&x + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right \| \end{gathered}$ $\boxed{ \rm{ OP \: R_1 \: \longmapsto \: R_1 + R_2 + R_3}}$ $\rm \: = \: \: \: \begin{gathered}\sf \left \| \begin{array}{ccc}2(x + y + z)&2(x + y + z)&x + y + z\\z + x&y + z& y\\x + y& z + x& z\end{array}\right \| \end{gathered}$ Take out x + y + z COMMON from Row 1, we get $\rm \: = \: (x + y + z)\begin{gathered}\sf \left \| \begin{array}{ccc}2&2&1\\z + x&y + z& y\\x + y& z + x& z\end{array}\right \| \end{gathered}$ $\boxed{ \rm{ OP \: C_1 \: \longmapsto \: C_1 - 2C_3}}$ $\rm \: = \: (x + y + z)\begin{gathered}\sf \left \| \begin{array}{ccc}0&2&1\\z + x - 2y&y + z& y\\x + y - 2z& z + x& z\end{array}\right \| \end{gathered}$ $\boxed{ \rm{ OP \: C_2 \: \longmapsto \: C_2 - 2C_3}}$ $\rm \: = \: (x + y + z)\begin{gathered}\sf \left \| \begin{array}{ccc}0&0&1\\z + x - 2y&z - y& y\\x + y - 2z& x - z& z\end{array}\right \| \end{gathered}$ On expanding along Row 1, we get $\rm \: = \: (x + y + z)\bigg((z + x - 2y)(x - z) - (x + y - 2z)(z - y)\bigg)$ $\rm \:=(x + y + z)\bigg( {x}^{2} - {z}^{2} - 2xy + 2yz - xz + xy - yz + {y}^{2} + {2z}^{2} - 2yz\bigg)$ $\rm \:=(x + y + z)\bigg( {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - zx \bigg)$ $\rm \: = \: \: {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$ Hence, $\rm \: \begin{gathered}\sf \left \| \begin{array}{ccc}y + z&x + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right \| \end{gathered} = {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$ Additional Information :- 1. The determinant VALUE remains unaltered if rows and columns are interchanged. 2. The determinant value is 0, if two rows or columns are identical. 3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged. 4. The determinant value remains unaltered if rows or columns are added or subtracted.

Answer»

$\large\underline{\sf{Solution-}}$

GIVEN determinant is TO PROVE

$\rm \: \begin{gathered}\sf \left | \begin{array}{ccc}y + z&<klux>X</klux> + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right | \end{gathered} = {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$

Consider LHS

$\rm \: = \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}y + z&x + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right | \end{gathered}$

$\boxed{ \rm{ OP \: R_1 \: \longmapsto \: R_1 + R_2 + R_3}}$

$\rm \: = \: \: \: \begin{gathered}\sf \left | \begin{array}{ccc}2(x + y + z)&2(x + y + z)&x + y + z\\z + x&y + z& y\\x + y& z + x& z\end{array}\right | \end{gathered}$

Take out x + y + z COMMON from Row 1, we get

$\rm \: = \: (x + y + z)\begin{gathered}\sf \left | \begin{array}{ccc}2&2&1\\z + x&y + z& y\\x + y& z + x& z\end{array}\right | \end{gathered}$

$\boxed{ \rm{ OP \: C_1 \: \longmapsto \: C_1 - 2C_3}}$

$\rm \: = \: (x + y + z)\begin{gathered}\sf \left | \begin{array}{ccc}0&2&1\\z + x - 2y&y + z& y\\x + y - 2z& z + x& z\end{array}\right | \end{gathered}$

$\boxed{ \rm{ OP \: C_2 \: \longmapsto \: C_2 - 2C_3}}$

$\rm \: = \: (x + y + z)\begin{gathered}\sf \left | \begin{array}{ccc}0&0&1\\z + x - 2y&z - y& y\\x + y - 2z& x - z& z\end{array}\right | \end{gathered}$

On expanding along Row 1, we get

$\rm \: = \: (x + y + z)\bigg((z + x - 2y)(x - z) - (x + y - 2z)(z - y)\bigg)$

$\rm \:=(x + y + z)\bigg( {x}^{2} - {z}^{2} - 2xy + 2yz - xz + xy - yz + {y}^{2} + {2z}^{2} - 2yz\bigg)$

$\rm \:=(x + y + z)\bigg( {x}^{2} + {y}^{2} + {z}^{2} - xy - yz - zx \bigg)$

$\rm \: = \: \: {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$

Hence,

$\rm \: \begin{gathered}\sf \left | \begin{array}{ccc}y + z&x + y&x\\z + x&y + z& y\\x + y& z + x& z\end{array}\right | \end{gathered} = {x}^{3} + {y}^{3} + {z}^{3} - 3xyz$

Additional Information :-

1. The determinant VALUE remains unaltered if rows and columns are interchanged.

2. The determinant value is 0, if two rows or columns are identical.

3. The determinant value is multiplied by - 1, if successive rows or columns are interchanged.

4. The determinant value remains unaltered if rows or columns are added or subtracted.