Prove that $\left|\begin{array}{ccc}a^{2} & b c & a c+c^{2} \\ a^{2}+a b & b^{2} & a c \\ a b & b^{2}+b c & c^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$
Prove that $\left|\begin{array}{ccc}a^{2} & b c & a c+c^{2} \\ a^{2}+a b & b^{2} & a c \\ a b & b^{2}+b c & c^{2}\end{array}\right|=4 a^{2} b^{2} c^{2}$
$\Delta=\left|\begin{array}{ccc}a^{2} & b c & a c+c^{2} \\ a^{2}+a b & b^{2} & a c \\ a b & b^{2}+b c & c^{2}\end{array}\right|$
Taking out common factors $a, b$ and $c$ from $C_{1}, C_{2},$ and $C_{3},$ we have:
$\Delta=a b c\left|\begin{array}{ccc}a & c & a+c \\ a+b & b & a \\ b & b+c & c\end{array}\right|$
Applying $R_{2} \rightarrow R_{2}-R_{1}$ and $R_{3} \rightarrow R_{3}-R_{1},$ we have:
$\Delta=a b c\left|\begin{array}{ccc}a & c & a+c \\ b & b-c & -c \\ b-a & b & -a\end{array}\right|$
Applying $R_{2} \rightarrow R_{2}+R_{1},$ we have:
$\Delta=a b c\left|\begin{array}{ccc}a & c & a+c \\ a+b & b & a \\ b-a & b & -a\end{array}\right|$
Applying $R_{3} \rightarrow R_{3}+R_{2},$ we have:
$\Delta=a b c\left|\begin{array}{ccc}a & c & a+c \\ a+b & b & a \\ 2 b & 2 b & 0\end{array}\right|$
$=2 a b^{2} c\left|\begin{array}{ccc}a & c & a+c \\ a+b & b & a \\ 1 & 1 & 0\end{array}\right|$
Applying $C_{2} \rightarrow C_{2}-C_{1},$ we have:
$\Delta=2 a b^{2} c\left|\begin{array}{ccc}a & c-a & a+c \\ a+b & -a & a \\ 1 & 0 & 0\end{array}\right|$
Expanding along $R_{3},$ we have:
$\Delta=2 a b^{2} c[a(c-a)+a(a+c)]$
$=2 a b^{2} c\left[a c-a^{2}+a^{2}+a c\right]$
$=2 a b^{2} c(2 a c)$
$=4 a^{2} b^{2} c^{2}$
Hence, the given result is proved.
Similar Questions
Evaluate $\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \operatorname{csin} \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$
Evaluate $\left|\begin{array}{ccc}\cos \alpha \cos \beta & \cos \alpha \operatorname{csin} \beta & -\sin \alpha \\ -\sin \beta & \cos \beta & 0 \\ \sin \alpha \cos \beta & \sin \alpha \sin \beta & \cos \alpha\end{array}\right|$
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^{-1}=\operatorname{adj}(\operatorname{adj} M )$, then which of the following statement is/are $ALWAYS TRUE$ ?
$(A)$ $M=I$ $(B)$ $\operatorname{det} M =1$ $(C)$ $M ^2= I$ $(D)$ $(\operatorname{adj} M)^2=I$
Let $M$ be a $3 \times 3$ invertible matrix with real entries and let $I$ denote the $3 \times 3$ identity matrix. If $M ^{-1}=\operatorname{adj}(\operatorname{adj} M )$, then which of the following statement is/are $ALWAYS TRUE$ ?
$(A)$ $M=I$ $(B)$ $\operatorname{det} M =1$ $(C)$ $M ^2= I$ $(D)$ $(\operatorname{adj} M)^2=I$
- [IIT 2020]
The value of $\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|$ is
The value of $\left|\begin{array}{lll}(a+1)(a+2) & a+2 & 1 \\ (a+2)(a+3) & a+3 & 1 \\ (a+3)(a+4) & a+4 & 1\end{array}\right|$ is
- [JEE MAIN 2021]
Using properties of determinants, prove that:
$\left|\begin{array}{lll}x & x^{2} & 1+p x^{3} \\ y & y^{2} & 1+p y^{3} \\ z & z^{2} & 1+p z^{3}\end{array}\right|=(1+p x y z)(x-y)(y-z)(z-x),$ where $p$ is any scalar.
Using properties of determinants, prove that:
$\left|\begin{array}{lll}x & x^{2} & 1+p x^{3} \\ y & y^{2} & 1+p y^{3} \\ z & z^{2} & 1+p z^{3}\end{array}\right|=(1+p x y z)(x-y)(y-z)(z-x),$ where $p$ is any scalar.