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Home  >>  CBSE XII  >>  Math  >>  Determinants
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Examine the consistency of the system of equations:\[\] \[\quad x + 3y = 5 \] \[\quad 2x + 6y = 8\]

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Toolbox:
  • (i)A matrix is said to be invertible if inverse exists.
  • (ii)If A is a non singular matrix such that
  • AX=B
  • then X=$A^{-1}B$
  • using this we can solve system of equations,which has unique solutions.
  • (iii)If A is a singular matrix,then |A|=0.
  • In this case we calculate (adj A)B.
  • If (adj A)B$\neq 0$,then there is no solution and the system is said the system is said to be in consistent.
The given system can be written in the form AX=B,
 
$A=\begin{bmatrix}1 & 3\\2 & 6\end{bmatrix}\;X=\begin{bmatrix}x\\y\end{bmatrix}\;B=\begin{bmatrix}5\\8\end{bmatrix}$
 
Hence $\begin{bmatrix}1 & 3\\2 & 6\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}=\begin{bmatrix}5\\8\end{bmatrix}$
 
Now let us find the value of the determinant
 
|A|=6-6=0
 
Hence it is a singular matrix.
 
Now let us calculate (adj A)
 
(adj A) can be calculated by interchanging the elements $a_{11}$ and $a_{22}$ and then by changing symbols of $a_{12}$ and $a_{21}$
 
Therefore $(adj A)=\begin{bmatrix}6 & -3\\-2 & 1\end{bmatrix}$
 
$(adj A)B=\begin{bmatrix}6 & -3\\-2 & 1\end{bmatrix}\begin{bmatrix}5\\8\end{bmatrix}=\begin{bmatrix}30 -24\\-10+8\end{bmatrix}=\begin{bmatrix}6\\-2\end{bmatrix}\neq 0$
 
Matrix multiplication can be obtained by multiplying the rows with the column.
 
Since $(adj A)B\neq 0$,the solution of the given system does not exist.hence it is consistent.

 

answered Feb 26, 2013 by sreemathi.v
edited Feb 28, 2013 by vijayalakshmi_ramakrishnans
 

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