Question

determinant properties question ​

Answer 1

Prove that

$\rm \:\: \: \begin{gathered}\sf \left | \begin{array}{ccc}1 + a&1&1\\1&1 + b&1\\1&1&1 + c\end{array}\right | \end{gathered} = abc + bc + ca + ab$

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

Consider,

$\rm \:\: \: \begin{gathered}\sf \left | \begin{array}{ccc}1 + a&1&1\\1&1 + b&1\\1&1&1 + c\end{array}\right | \end{gathered}$

Taking a, b, c common from Row 1, Row 2 and Row 3, we get

$\rm \:\: = abc \: \begin{gathered}\sf \left | \begin{array}{ccc}1 + \dfrac{1}{a} &\dfrac{1}{a} &\dfrac{1}{a} \\\dfrac{1}{b}&1 + \dfrac{1}{b} &\dfrac{1}{b} \\\dfrac{1}{c} &\dfrac{1}{c} &1 + \dfrac{1}{c} \end{array}\right | \end{gathered}$

$\rm :\longmapsto\:OP \: R_1 \: \to \: R_1 + R_2 + R_3$

$\rm \:\: = abc \: \begin{gathered}\sf \left | \begin{array}{ccc}1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} &1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} &1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c} \\\dfrac{1}{b}&1 + \dfrac{1}{b} &\dfrac{1}{b} \\\dfrac{1}{c} &\dfrac{1}{c} &1 + \dfrac{1}{c} \end{array}\right | \end{gathered}$

$\rm \:= abc(1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}) \: \begin{gathered}\sf \left | \begin{array}{ccc}1 &1&1 \\\dfrac{1}{b}&1 + \dfrac{1}{b} &\dfrac{1}{b} \\\dfrac{1}{c} &\dfrac{1}{c} &1 + \dfrac{1}{c} \end{array}\right | \end{gathered}$

$\rm :\longmapsto\:OP \: C_2 \to \: C_2 - C_1 \: and \: OP \: C_3 \to \: C_3 - C_1$

$\rm \: = abc(1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c})\: \begin{gathered}\sf \left | \begin{array}{ccc}1&0&0\\\dfrac{1}{b}&1&0\\\dfrac{1}{c}&0&1\end{array}\right | \end{gathered}$

On expanding the determinant along Row1, we get

$\rm \: = \: \: abc(1 + \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}) \times 1$

$\rm \: = \: \: abc + bc + ca + ab$

${\boxed{\boxed{\bf{Hence, Proved}}}}$

Additional Information :-

1. The determinant remains unaltered if rows or columns are interchanged.

2. The determinant value is multiplied by (- 1) if successive rows or columns are interchanged.

3. The determinant value remains unchanged if rows or columns are added.

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

5. The determinant value is 0, if one row or column is zero.