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

Q : 5        Examine the consistency of the system of equations.

                \small 3x-y-2z=2

                \small 2y-z=-1

                \small 3x-5y=3

Answers (1)

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We have given the system of equations:

                                  \small 3x-y-2z=2

                                    \small 2y-z=-1

                                     \small 3x-5y=3

The given system of equations can be written in the form of matrix; 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}  and B = \begin{bmatrix} 2\\-1 \\ 3 \end{bmatrix}.

So, we want to check for the consistency of the equations;

|A| = 3(0-5) -(-1)(0+3)-2(0-6)

= -15 +3+12 = 0 

Therefore matrix A is a singular matrix.

So, we will then check (adjA)B,

(adjA) = \begin{bmatrix} -5 &10 &5 \\ -3& 6 & 3\\ -6& 12 & 6 \end{bmatrix}

\therefore (adjA)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} = \begin{bmatrix} -5\\-3 \\ -6 \end{bmatrix} \neq 0

As, (adjA)B is non-zero thus the solution of the given system of the equation does not exist. Hence, the given system of equations is inconsistent.

Posted by

Divya Prakash Singh

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