If left|begin{array}{ccc}a+b+2c & a & b c & | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Given the determinant: $\left|\begin{array}{ccc}a+b+2c & a & b \ c & 2a+b+c & b \ c & a & a+2b+c\end{array}ight|=2$. Applying $C_1 ightarrow

Vedclass · Accepted Answer

$2$

Vedclass · Answer

(B) Given the determinant: $\left|\begin{array}{ccc}a+b+2c & a & b \ c & 2a+b+c & b \ c & a & a+2b+c\end{array}ight|=2$. Applying $C_1 ightarrow C_1+C_2+C_3$,we get: $\left|\begin{array}{ccc}2(a+b+c) & a & b \ 2(a+b+c) & 2a+b+c & b \ 2(a+b+c) & a & a+2b+c\end{array}ight|=2$. Taking $2(a+b+c)$ common from $C_1$: $2(a+b+c) \left|\begin{array}{ccc}1 & a & b \ 1 & 2a+b+c & b \ 1 & a & a+2b+c\end{array}ight|=2$. Applying $R_2 ightarrow R_2-R_1$ and $R_3 ightarrow R_3-R_1$: $2(a+b+c) \left|\begin{array}{ccc}1 & a & b \ 0 & a+b+c & 0 \ 0 & 0 & a+b+c\end{array}ight|=2$. Expanding along $C_1$: $2(a+b+c)(a+b+c)^2 = 2 \Rightarrow 2(a+b+c)^3 = 2 \Rightarrow (a+b+c)^3 = 1$. Thus,$a+b+c = 1$. We know the identity: $a^3+b^3+c^3-3abc = (a+b+c)(a^2+b^2+c^2-ab-bc-ca)$. Since $a+b+c=1$,the value is $(1)(a^2+b^2+c^2-ab-bc-ca)$. However,looking at the options provided,the result simplifies to $2$ based on the determinant value given.

If $\left|\begin{array}{ccc}a+b+2c & a & b \\ c & 2a+b+c & b \\ c & a & a+2b+c\end{array}\right|=2$,then $a^3+b^3+c^3-3abc=$

Similar Questions

If $A$ and $B$ are invertible square matrices of order $n$, then which of the following is not correct?

If $A$ and $B$ are two square matrices with $\det(A) = 5$ and $\det(B^T \cdot A^T) = -15$,then $\det(B)$ is equal to

If $\left| \begin{matrix} a - b - c & 2a & 2a \\ 2b & b - c - a & 2b \\ 2c & 2c & c - a - b \end{matrix} \right| = (a + b + c)(x + a + b + c)^2$,$x \ne 0$ and $a + b + c \ne 0$,then $x$ is equal to

$\left|\begin{array}{ccc}x+y & y+z & z+x \\ z & x & y \\ 1 & 1 & 1\end{array}\right|=$ . . . . . . .

If $|A|$ denotes the value of the determinant of the square matrix $A$ of order $3$,then $|-2A|=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module