❖ Using Matrix inverse to solve two Linear Systems.

We are using Matrix inverse to solve two Linear Systems.

First:

You need to write an Augmented Matrix containing the original Matrix and the Identity Matrix,

Then:

You need to convert the original Matrix into the Identity Matrix using Elementary Row Operations

(Apply Reduced Row Echelon Form (RREF) for the left side of the Augmented Matrix).

The Identity Matrix will convert into the inverse of the original Matrix as long as you apply the same Elementary Row Operations for the Augmented Matrix.

Since \(A^{-1}\) exists, then, we can multiply both sides of the Linear Systems \(Ax=b\) and \(Ax=c\) by \(A^{-1}\) from the left.

Simply, we do:

\(x=A^{-1} b\) for linear system \(Ax=b\),

and

\(x=A^{-1} c\) for linear system \(Ax=c\),

where \(b\) and \(c\) are the result vectors.

Note:

If \(A\) inverse exists, then \(A A^{-1} = A^{-1} A =\) Identity Matrix

If \(A\) is not a Square Matrix, then the inverse of Matrix \(A\) is DNE.

The Square Matrix is a Matrix with

the number of Rows \(=\) the number of Columns.

❖ The number of Rows and Columns that a Matrix has is called its Size, Order, or Dimension.