Question
Without expanding at any stage, find the value of $\left|\begin{array}{ccc}a & b & c \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right|.$
✓
Answer
Let, $\Delta=\left|\begin{array}{ccc}a & b & c \\ a+2 x & b+2 y & c+2 z \\ x & y & z\end{array}\right|$
Applying $R_1 \Rightarrow R_1+2 R_3$, we get
$\Delta=\left|\begin{array}{ccc}a+2 x & b+2 y & c+2 z \\a+2 x & b+2 y & c+2 z \\x & y & z\end{array}\right|=0$
Since, $\text{R}_1$ and $\text{R}_2$ are identical, therefore value of $\Delta$ is zero.
Applying $R_1 \Rightarrow R_1+2 R_3$, we get
$\Delta=\left|\begin{array}{ccc}a+2 x & b+2 y & c+2 z \\a+2 x & b+2 y & c+2 z \\x & y & z\end{array}\right|=0$
Since, $\text{R}_1$ and $\text{R}_2$ are identical, therefore value of $\Delta$ is zero.
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Using matrix method, solve the following system of equations:
$
\begin{array}{l}
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\end{array}
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\begin{array}{l}
x-2 y+3 z=6 \\
x+4 y+z=12 \\
x-3 y+2 z=1
\end{array}
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