यदि \(\rm \Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}\) है, तो \(\rm \begin{vmatrix} 3d + 5g & 4a + 7g & 6g\\ 3e + 5h & 4b + 7h & 6h\\ 3f + 5i & 4c + 7i & 6i \end{vmatrix}\) किसके बराबर होगा?

Concept:: 

• The interchange of any two rows (or columns) of the determinant changes its sign.
• The determinant remains unaltered if its rows are changed into columns and the columns into rows.

गणना:

\(\Delta = \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix}\)

\(A = \begin{vmatrix} 3d + 5g & 4a + 7g & 6g\\ 3e + 5h & 4b + 7h & 6h\\ 3f + 5i & 4c + 7i & 6i \end{vmatrix}\)

\(⇒ A = 6\begin{vmatrix} 3d + 5g & 4a + 7g & g\\ 3e + 5h & 4b + 7h & h\\ 3f + 5i & 4c + 7i & i \end{vmatrix}\)

C1 → C1 - 5C3, C2 → C2 - 7C3

\(A = 6\begin{vmatrix} 3d & 4a & g\\ 3e & 4b & h\\ 3f & 4c& i \end{vmatrix}\)

\(A =6\times 3\times4 \begin{vmatrix} d & a & g\\ e & b & h\\ f & c& i \end{vmatrix}\)

Using the property of determinants

\(A =-72 \begin{vmatrix} a & b & c\\ d & e & f\\ g & h& i \end{vmatrix}\)

∴  A = -72Δ