**3.7.3.1 Linear equation in matrix notation:** take the following system of linear equation;

a11*x*+a12*x*+....+a1_{n}*x*n

a21*x*+a22*x*+....+a2_{n}*x*_{n} = b2

........ .... .... ... ... ... ....

... ... ... ... .... .... .... ....

a_{n}1*x*+a_{n}2*x*+....+a_{nn}*x*_{n} = b_{n}

and,

The above system can also be written in the form;

AX = B

It can be rearranged as;

Let X = X1 and X = X2 be two solutions of AX = B

So,

AX1 = B and AX2 = B

Therefore,

AX1 = AX2

Since A being invertible, by cancellation law, we have;

X1 = X2

Hence the given system of equation has unique solution.

**Criterion for a given system of equation to have unique solution:**

i) If |A| ≠ 0, then the system is consistent and has a unique solution and is given by

X = A^{-1}B

- ii) If |A| = 0 and (adjA) B ≠ 0, then the system is inconsistent.
- iii) If |A| = 0 and (adjA) B = 0 then the system is consistent and has infinitely many solution.

Example: Using matrix method, solve the system of linear equation;

5*x* + 3y + 2 = 0

2*x* + 3y + 3= 0

Solution: the given set equations are

5*x* + 7y + 2 = 0

2*x* + 3y + 3= 0

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