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Home  >>  CBSE XII  >>  Math  >>  Determinants
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Examine the consistency of the system of equations:\[\] \[\quad x + y + z = 1 \] \[\quad 2x + 3y +2z = 2\] \[\quad ax + ay +2az = 4\]

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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.
  • If (adj A)B$\neq 0$,then there is no solution and the system is said the system is said to be in consistent.
This can be written in the form AX=B
 
$\begin{bmatrix}1 & 1& 1\\2 & 3& 2\\a & a &2a\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}1\\2\\4\end{bmatrix}$
 
Where $A=\begin{bmatrix}1 & 1& 1\\2 & 3& 2\\a & a&2a\end{bmatrix}\;X=\begin{bmatrix}x\\y\\z\end{bmatrix}\;B=\begin{bmatrix}1\\2\\4\end{bmatrix}$
 
Let us evaluate the value of determinant |A| by expanding along $R_1$
 
$|A|=1(3\times 2a-2\times a)-1(2\times 2a-2\times a)+1(2\times a-3a)$
 
$\;\;\;=6a-2a-4a+2a+2a-3a$
 
$\;\;\;=a\neq 0$
 
Therefore A is non singular.
 
Hence $A^{-1}$ exists.
 
Hence the given system of equation is consistent.

 

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

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