# Applications of Determinants and Matrices

#### Topics

• Consistent System
• Inconsistent System
• Solution of a system of linear equations using the inverse of a matrix

## Notes

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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Solution of System of Linear Equations by Inversion Method Part 1 [00:06:28]
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