ISC (Arts) Class 12CISCE
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Applications of Determinants and Matrices

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Consistent system: A system of equations is said to be consistent if its solution (one or more) exists. 
Inconsistent system: A system of equations is said to be inconsistent if its solution does not exist.  

Solution of system of linear equations using inverse of a matrix:
Consider the system of equations
`a_1 x + b_1 y + c_1 z = d_1` 
`a_2 x + b_2 y + c_2 z = d_2` 
`a _3 x + b_3 y + c_3 z = d_3`

Let A =` [(a_1,b_1,c_1),(a_1,b_2,c_2),(a_3,b_3,c_3)] , X = [x,y,z]` and

B =` [(d_1),(d_2),(d_3)]`
Then, the system of equations can be written as, AX = B, i.e.,
`[(a_1,b_1,c_1),(a_2,b_2,c_2),(a_3,b_3,c_3)] [(x),(y),(z)] = [(d_1),(d_2),(d_3)]`

Case I:  If A is a nonsingular matrix, then its inverse exists. Now
AX = B 
or `A^(–1)` (AX) = `A^(–1)` B       (premultiplying by A–1) 
or `(A^(–1)A)` X = `A^(–1)` B       (by associative property) 
or I X = `A^(–1)` B 
or X = `A^(–1)` B 
This matrix equation provides unique solution for the given system of equations as inverse of a matrix is unique. This method of solving system of equations is known as Matrix Method.
Case II:   If A is a singular matrix, then |A| = 0. 
In this case, we calculate (adj A) B. 
If (adj A) B ≠ O, (O being zero matrix), then solution does not exist and the system of equations is called inconsistent. 
If (adj A) B = O, then system may be either consistent or inconsistent according as the system have either infinitely many solutions or no solution.

Video link : https://youtu.be/lBnSOCO-9EI

Shaalaa.com | Solution of System of Linear Equations by Inversion Method Part 1

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