Using properties of determinants in Exercises 11 to 15 , prove that:
$$
\left|\begin{array}{ccc}
3 a & -a+b & -a+c \\
-b+a & 3 b & -b+c \\
-c+a & -c+b & 3 c
\end{array}\right|=3(a+b+c)(a b+b c+c a)
$$

Answer

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Algebra

NCERT Class 12 Part 1 - Math

Chapter 4

Determinants

Section 7

Applications of Determinants and Matrices

Discussion

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Video Transcript

We have all the number 13 in this we have to okay in this we have to prove using properties of determinant that three A minus B plus a minus C plus a minus a plus B. Three B minus C plus b minus a plus c minus B plus c. Three C determinant equal to three A plus B plus C. Mhm. Into A B Place Bc place. See so left inside. Okay. Honey lived in sale. We have to operate minus a minus A. So called him one. Call them one place called him to place column three. So this will be simply A plus B plus c minus A plus B. Mine's A plus C. A place B plus C mm Three B minus B plus C. And so A place B plus C minus C plus B. And crazy. Now let us take A plus B plus C. As common. A plus B plus C. Common. We are left with +111 minus A plus B. Mine is able to see three B minus B plus c minus C plus b Tracy. Okay you know this is 111 so we have to make it zero zero and one or 100 That is two zeros. So let us operate our three our three minus R. One and R. Two our two minus R. One. We'll be getting a place B plus C. Yeah okay this will be 100 Mine's a plus B minus a placebo. Women as it is three B plus a minus B. So simply we should right here to be yeah this will become three be managed to. Okay so A plus okay a place to be and then a minus B and then here it will be a minus C. And here it will be to see place A Okay now we'll be operating just like this and this. So this will be a place be policy into A plus to be into a place to see minus a minus C. A minus B. So A plus B plus C into let us multiply is choir plus two maybe plus four Bc minus a square plus A B bless A C minus busy also is quite an exquisite cancel out we'll be having a place B policy into this is to A B. Okay to A B plus 83. Abe bless three B C for busy months Okay let's three A C. For me it's either to A C. I left to a simplicity three easy so this this is yeah if you take three as comment will be three A place people see A B plus B C plus a C. They should be the answer thank you. And this is our right inside story thank

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