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Question

Examine the consistency of the system of equations$$3x-y-2z=2, 2y-z=-1, 3x-5y=3$$


Solution

Given system of equations
$$3x-y-2z=2$$
$$2y-z=-1$$
$$3x-5y=3$$
This can be written as 
$$AX=B$$
where $$A=\begin{bmatrix} 3 & -1 &-2 \\ 0 & 2 & -1\\ 3 &-5 & 0 \end{bmatrix}, X=\begin{bmatrix} x \\ y\\z \end{bmatrix}, B=\begin{bmatrix} 2 \\ -1\\ 3\end{bmatrix}$$

Here, $$|A|=3(0-5)+1(0+3)-2(0-6)$$
$$\Rightarrow |A|=0$$
Since, $$|A|= 0$$
Hence, the system of equations has either infinitely many solutions (consistent) or no solution (inconsistent).

We need to find $$(adj A)B$$
$$C_{11}=(-1)^{1+1} \begin{vmatrix} 2 & -1 \\ -5 & 0 \end{vmatrix}$$
$$\Rightarrow C_{11}=0-5 =-5$$

$$C_{12}=(-1)^{1+2} \begin{vmatrix} 0 & -1 \\ 3 & 0 \end{vmatrix}$$
$$\Rightarrow C_{12}=-(0+3) =-3$$

$$C_{13}=(-1)^{1+3} \begin{vmatrix} 0 & 2 \\ 3 & -5 \end{vmatrix}$$
$$\Rightarrow C_{13}=0-6 =-6$$

$$C_{21}=(-1)^{2+1} \begin{vmatrix} -1 & -2 \\ -5 & 0 \end{vmatrix}$$
$$\Rightarrow C_{21}=-(0-10) =10$$

$$C_{22}=(-1)^{2+2} \begin{vmatrix} 3 & -2 \\ 3 & 0 \end{vmatrix}$$
$$\Rightarrow C_{22}=0+6 =6$$

$$C_{23}=(-1)^{2+3} \begin{vmatrix} 3 & -1 \\ 3 & -5 \end{vmatrix}$$
$$\Rightarrow C_{23}=-(-15+3) =12$$

$$C_{31}=(-1)^{3+1} \begin{vmatrix} -1 & -2 \\ 2 & -1 \end{vmatrix}$$
$$\Rightarrow C_{31}=1+4 =5$$

$$C_{32}=(-1)^{3+2} \begin{vmatrix} 3 & -2 \\ 0 & -1 \end{vmatrix}$$
$$\Rightarrow C_{32}=-(-3-0) =3$$

$$C_{33}=(-1)^{3+3} \begin{vmatrix} 3 & -1 \\ 0 & 2 \end{vmatrix}$$
$$\Rightarrow C_{33}=6-0=6$$

Hence, the co-factor matrix is $$C=\begin{bmatrix} -5 & -3 & -6 \\ 10 & 6 & 12 \\ 5 & 3 & 6 \end{bmatrix}$$

$$\Rightarrow adj A= C^{T}=\begin{bmatrix} -5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6 \end{bmatrix}$$

Now, $$(adj A)B=\begin{bmatrix} -5 & 10 & 5 \\ -3 & 6 & 3 \\ -6 & 12 & 6 \end{bmatrix}\begin{bmatrix} 2 \\ -1 \\ 3 \end{bmatrix}$$

$$=\begin{bmatrix} -10-10+15 \\ -6-6+9 \\ -12-12+18 \end{bmatrix}$$

$$\Rightarrow (adj A)B=\begin{bmatrix} -5 \\ -3 \\ -6 \end{bmatrix}$$
Since, $$(adj A)B \ne O$$
Hence, the system of equations is inconsistent.

Mathematics
NCERT
Standard XII

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