Question

Using the properties of determinants, prove that

1 \\\ 1 + 3y \\\ 1 \\\ \end{gathered} \right.$$ $\begin{gathered} 1 \\\ 1 \\\ 1 + 3z \\\ \end{gathered} $ $$\left. \begin{gathered} 1 + 3x \\\ 1 \\\ 1 \\\ \end{gathered} \right| = 9(3xyz + xy + yz + xz)$$

Answer 1

Hint: In these types of questions remember to use the row -column transformation method for example ${R_3} \to {R_3} - {R_1}$, ${R_2} \to {R_2} - {R_1}$ to solve the question. After doing this row-column transformation, then we just need to find the determinants. We will get the required answer.

Complete step-by-step answer:
According to the given information the given determinant is \left| \begin{gathered} 1 \\\ 1 + 3y \\\ 1 \\\ \end{gathered} \right.$$$\begin{gathered} 1 \\\ 1 \\\ 1 + 3z \\\ \end{gathered} $$$\left. \begin{gathered} 1 + 3x \\\ 1 \\\ 1 \\\ \end{gathered} \right|

Applying ${R_2} \to {R_2} - {R_1}$
So the determinant becomes $\left| \begin{gathered} 1 \\\ 3y \\\ 1 \\\ \end{gathered} \right.$ $\begin{gathered} 1 \\\ 0 \\\ 1 + 3z \\\ \end{gathered}$ $\left. \begin{gathered} 1 + 3x \\\ \- 3x \\\ 1 \\\ \end{gathered} \right|$

Now applying ${R_3} \to {R_3} - {R_1}$

1 \\\ 3y \\\ 0 \\\ \end{gathered} \right.$$ $\begin{gathered} 1 \\\ 0 \\\ 3z \\\ \end{gathered} $ $$\left. \begin{gathered} 1 + 3x \\\ \- 3x \\\ \- 3x \\\ \end{gathered} \right|$$ Now, solving the determinant form along ${C_1}$ $$ \Rightarrow $$1(3x) (3z) – 3y (-3x – 3z – 9xz) = 27xyz+9xy+9yz+9xz $$ \Rightarrow $$9(3xyz+xy+yz+zx) since LHS = RHS Hence proved. Note: In these types of questions first use the row-column transformation method for example ${R_2} \to {R_2} - {R_1}$, ${R_3} \to {R_3} - {R_1}$ to change the matrix such that when we solve the determinant of matrix simplify the result until the LHS becomes equal to RHS and we will get the result we required.