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Solve by matrix inversion method following system of linear equation:$\;2x\;-\;y\;=7\;,\;3x\;-2y\;=11$

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Toolbox:
  • Let A be a square matrix of order 2. $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then $ adj\: A = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ ( shortcut for finding the adjoint )
  • The inverse of a non-singular square matrix A is given by $ A^{-1} = \frac{1}{|A|} adj\: A.$ A non-singular matrix is one whose determinant value is nonzero.
  • A system of x linear non-homogeneous equations in x unknowns given by
  • $ a_{11}x_1 +a_{12}x_2 \dotsc + a_{1n}x_n = b_1$
  • $ a_{22}x_1 +a_{22}x_2 \dotsc + a_{2n}x_n = b_2$
  • $\: \: \: \: \vdots \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \vdots $
  • $ a_{n1}x_1 +a_{n2}x_2 \dotsc + a_{nn}x_n = b_n$ can be written as $ AX = B$ in matrix form where $ A = [a_{ij}], x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$ and $ B = \begin{bmatrix} b_1 \\ b_2 \\ \vdots \\ b_n \end{bmatrix}$ If A is non-singular $ ( |A| \neq 0 )$ the inverse $A^{-1}$ exists and the system can be solved by matrix inversion $ X = A^{-1}B$
Step 1
The matrix equation is $ AX = B$ with $ A = \begin{bmatrix} 2 & -1 \\ 3 & -2 \end{bmatrix}, X = \begin{bmatrix} x \\ y \end{bmatrix}, B = \begin{bmatrix} 7 \\ 11 \end{bmatrix} $
$ |A| = -4+3 = -1 \neq 0 $ The system can be solved by matrix inversion. $ X = A^{-1} B$
Step 2
To find $ A^{-1} = \large\frac{1}{|A|} adj\: A$
$ adj\: A = \begin{bmatrix} -2 & 1 \\ -3 & 2 \end{bmatrix}$
$ A^{-1} = \large\frac{1}{-1} \begin{bmatrix} -2 & 1 \\ -3 & 2 \end{bmatrix} = \begin{bmatrix} 2 & -1 \\ 3 & -2 \end{bmatrix}$
Step 3
$ X = A^{-1}B = \begin{bmatrix} 2 & -1 \\ 3 & -2 \end{bmatrix} \begin{bmatrix} 7 \\ 11 \end{bmatrix} = \begin{bmatrix} 14-11 \\ 21-22 \end{bmatrix} = \begin{bmatrix} 3 \\ -1 \end{bmatrix}$
$ \therefore x = 3, y = -1$

 

answered May 24, 2013 by thanvigandhi_1
edited May 27, 2013 by thanvigandhi_1
 

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