A=-bcb2+bcc2+bca2+ac-acc2+aca2+abb2+ab-abMultiply R1 , R2 and R3 by a, b , and c respectively and divide the determinant by abc 1abc-abcab2+abcac2+abcba2+bac-acbbc2+bacca2+cabcb2+cab-cabTake a , b , and c from C1 , C2 and C3 abcabc-bcab+acac+abba+bc-acbc+baca+cbcb+ca-abApply R1→R1+R2+R3ab+bc+caab+bc+caab+bc+caba+bc-acbc+baca+cbcb+ca-abTaking ab+bc+ca common from R1 ab+bc+ca111ba+bc-acbc+baca+cbcb+ca-abApply C2→C2-C1 and C3→C3-C1ab+bc+ca100ba+bc-ac-ba-bc0ca+cb0-ab-ca-cbExpanding along R1ab+bc+ca1ab+bc+ca2=ab+bc+ca3=RHSHence Proved

Ques. No 4

|A| = |\begin{matrix} - bc & b^{2} + bc & c^{2} + bc \\ a^{2} + ac & - ac & c^{2} + ac \\ a^{2} + ab & b^{2} + ab & - ab \end{matrix}| Multiply R_{1}, R_{2} and R_{3} by a, b, and c respectively and divide the determinant by abc . \frac{1}{abc} |\begin{matrix} - abc & {ab}^{2} + abc & {ac}^{2} + abc \\ {ba}^{2} + bac & - acb & {bc}^{2} + bac \\ {ca}^{2} + cab & {cb}^{2} + cab & - cab \end{matrix}| Take a, b, and c from C_{1}, C_{2} and C_{3} . \frac{abc}{abc} |\begin{matrix} - bc & ab + ac & ac + ab \\ ba + bc & - ac & bc + ba \\ ca + cb & cb + ca & - ab \end{matrix}| Apply R_{1} \to R_{1} + R_{2} + R_{3} |\begin{matrix} (ab + bc + ca) & (ab + bc + ca) & (ab + bc + ca) \\ ba + bc & - ac & bc + ba \\ ca + cb & cb + ca & - ab \end{matrix}| Taking (ab + bc + ca) common from R_{1} (ab + bc + ca) |\begin{matrix} 1 & 1 & 1 \\ ba + bc & - ac & bc + ba \\ ca + cb & cb + ca & - ab \end{matrix}| Apply C_{2} \to C_{2} - C_{1} and C_{3} \to C_{3} - C_{1} (ab + bc + ca) |\begin{matrix} 1 & 0 & 0 \\ ba + bc & - ac - ba - bc & 0 \\ ca + cb & 0 & - ab - ca - cb \end{matrix}| Expanding along R_{1} (ab + bc + ca) [1 {(ab + bc + ca)}^{2}] = {(ab + bc + ca)}^{3} = RHS Hence Proved .

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