#### Explain solution RD Sharma class 12 chapter Solution of Simultaneous Linear Equation exercise 7.1 question 4 subquestion (iv) maths

Inconsistent

Given:

$4 x-5 y-2 z=2 \quad, \quad 5 x-4 y+2 z=-2 \quad, \quad 2 x+2 y+8 z=-1$

Hint:

Inconsistent means two or more equations that are impossible to solve based on using one set of values for variables.

Solution:

The given system of equations can be expressed as follows

$A X=B \\ Here, \\ A=\left[\begin{array}{ccc}4 & -5 & -2 \\ 5 & -4 & 2 \\ 2 & 2 & 8\end{array}\right] \quad, \quad X=\left[\begin{array}{c}x \\ y \\ z\end{array}\right] \quad and \quad B=\left[\begin{array}{c}2 \\ -2 \\ -1\end{array}\right] \\ Now,$

\begin{aligned} |A|=\left|\begin{array}{ccc} 4 & -5 & -2 \\ 5 & -4 & 2 \\ 2 & 2 & 8 \end{array}\right| &=4(-32-4)+5(40-4)-2(10+8) \\ &=-144+180-36 \\ &=0 \end{aligned}

$C_{i j}\; be\; the\; co-\! f\! actor\; o\! f\; the\; elements\; a_{i j}\; in \; A=\left[a_{i j}\right].\; Then,$

\begin{aligned} &C_{11}=-1^{1+1}\left|\begin{array}{cc} -4 & 2 \\ 2 & 8 \end{array}\right|=28 \quad, \quad C_{21}=-1^{2+1}\left|\begin{array}{cc} -5 & -2 \\ 2 & 8 \end{array}\right|=36 \\ &C_{12}=-1^{1+2}\left|\begin{array}{ll} 5 & 2 \\ 2 & 8 \end{array}\right|=-36 \quad, \quad C_{22}=-1^{2+2}\left|\begin{array}{cc} 4 & -2 \\ 2 & 8 \end{array}\right|=36 \end{aligned}

\begin{aligned} &C_{13}=-1^{1+3}\left|\begin{array}{cc} 5 & -4 \\ 2 & 2 \end{array}\right|=18 \quad, \quad C_{23}=-1^{2+3}\left|\begin{array}{cc} 4 & -5 \\ 2 & 2 \end{array}\right|=-18 \\ &C_{31}=-1^{3+1}\left|\begin{array}{cc} -5 & -2 \\ -4 & 2 \end{array}\right|=-18 \quad, \quad C_{32}=-1^{3+2}\left|\begin{array}{cc} 4 & -2 \\ 5 & 2 \end{array}\right|=-18 \\ &C_{33}=-1^{3+3}\left|\begin{array}{cc} 4 & -5 \\ 5 & -4 \end{array}\right|=9 \end{aligned}

\begin{aligned} (\text { adjA }) B &=\left[\begin{array}{ccc} 28 & 36 & -18 \\ -36 & 36 & -18 \\ 18 & -18 & 9 \end{array}\right]\left[\begin{array}{c} 2 \\ -2 \\ -1 \end{array}\right] \\ &=\left[\begin{array}{c} 56-72+18 \\ -72-72+18 \\ 36+36-9 \end{array}\right] \\ &=\left[\begin{array}{c} 2 \\ -126 \\ 63 \end{array}\right] \neq 0 \end{aligned}

Hence, the given system of equation is inconsistent.