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.
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