### Inverse

##### 2. Inverse matrix

By the equivalence between the composition of linear transformations and the multiplication of matrices, we define a matrix **B** to be the inverse of a matrix **A** if

**BA** = **I**, and **AB** = **I**,

where **I** denotes the identity matrix. We also denote **B** = **A**^{-1}, similar to the notation for inverse transformations.

Our study of the inverse matrix is based on the following result. A generalization of the result can be found in this earlier exercise and this earlier exercise. The converse of the result can be found in this exercise.

**AB** = **I** ⇒ `Ax` = **b** has solution for any **b**, and solution of `By` = **c** is unique.

Proof For any **b**, the following verifies that **x** = `Bb` is a solution of `Ax` = **b**:

`Ax` = **A**(`Bb`) = (`AB`)**b** = `Ib` = **b**.

As for the uniqueness for `By` = **c**, let **y** and `y'` be two solutions. Then

`By` = **c**, `By'` = **c** ⇒ `ABy` = `Ac`, `ABy'` = `Ac` ⇒ **y** = `Ac` = `y'`,

where **AB** = **I** is used in the last step.

Thus the two conditions in the definition of inverse matrices imply that `Ax` = **b**
must have a unique solution for any **b**. In fact from the proof we also know what the solution is.

**A** is invertible ⇒ For any **b**, `Ax` = **b** has
**x** = **A**^{-1}**b** as the unique solution.

Moreover, by the numerical consequences of
always existence and
uniqueness,
we have

An invertible matrix must be a square matrix.

Translated to linear transformations, we see that there is no invertible linear transformations between
euclidean spaces of different dimensions.

If there is an invertible linear transformation from **R**^{n}
to **R**^{m}, then `m` = `n`.

Example We gave the zero transformation as an example
of non-invertible linear transformation. Now by the size consideration, we know that the linear transformation

**T**(`x`_{1}, `x`_{2}, `x`_{3})
= (`x`_{1} + 2`x`_{2} + 3`x`_{3}, 4`x`_{1}
+ 5`x`_{2} + 6`x`_{3}): **R**^{3} → **R**^{2}

is not invertible. Correspondingly, the matrix

is not invertible.

For the square matrix

[ |
3 |
1 |
-1 |
], |

1 |
-1 |
1 |

2 |
-1 |
1 |

because the corresponding system does not
always have solutions, the matrix is not invertible.