#### 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.

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