Mathematics - Determinant Question with Solution | TestHub

Let the given determinant be $D$:

$$D=\left|\begin{array}{ccc}1+a& 1& 1\\ 1& 1+b& 1\\ 1& 1& 1+c\end{array}\right|$$

Factor out $abc$ from the determinant:

$$D=abc\left|\begin{array}{ccc}\frac{1}{a}+1& \frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}& 1+\frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|$$

Apply the column operation ${\mathrm{C}}_1\to {\mathrm{C}}_1+{\mathrm{C}}_2+{\mathrm{C}}_3$:

$$D=abc\left|\begin{array}{ccc}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1& \frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1& 1+\frac{1}{b}& \frac{1}{c}\\ \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|$$

Factor out the common term from the first column:

$$D=abc\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1\right)\left|\begin{array}{ccc}1& \frac{1}{b}& \frac{1}{c}\\ 1& 1+\frac{1}{b}& \frac{1}{c}\\ 1& \frac{1}{b}& 1+\frac{1}{c}\end{array}\right|$$