# Left, Right Inverses: Multiple Left Inverses and No Right Inverse?

• psholtz
In summary, having multiple left inverses for a linear transformation or matrix A does not necessarily mean that A has no right inverse. This is because if A has both left and right inverses, they must be equal to each other and to the inverse of A. Therefore, A is invertible and its left and right inverses are the same.
psholtz
Suppose we have a linear transformation/matrix A, which has multiple left inverses B1, B2, etc., such that, e,g,:

$$B_1 \cdot A = I$$

Can we conclude from this (i.e., from the fact that A has multiple left inverses) that A has no right inverse?

If so, why is this?

Suppose there were a right inverse, say, R. Multiplying the equation $B_1A= I$ on the right by R gives $(B_1A)R= IR$ so that $B_1(AR)= B_1= R$ But then doing the same with $B_2A$ leads to $B_2= R$.

In other words, if a matrix has both right and left inverses, then it is invertible and both right and left inverses are equal to its (unique) inverse.

## 1. What is a left inverse?

A left inverse is an element that, when multiplied to the left of another element, results in the identity element. In other words, if a left inverse exists for an element a, then a multiplied by its left inverse will equal the identity element.

## 2. How is a left inverse different from a right inverse?

A left inverse is an element that, when multiplied to the left of another element, results in the identity element. A right inverse, on the other hand, is an element that, when multiplied to the right of another element, results in the identity element. In some cases, an element may have both a left and right inverse, while in other cases it may only have one or neither.

## 3. Can an element have multiple left inverses?

Yes, it is possible for an element to have multiple left inverses. This can occur when the set of elements being considered is not closed under multiplication, meaning that the result of multiplying two elements in the set is not necessarily in the set. In this case, an element may have multiple left inverses, each of which may result in the identity element when multiplied to the left of the original element.

## 4. What happens if an element has no right inverse?

If an element has no right inverse, it means that no element exists that can be multiplied to the right of the original element to result in the identity element. This may be due to the set of elements being considered not being closed under multiplication.

## 5. How are left inverses used in mathematics and science?

Left inverses are used in various mathematical and scientific applications, including linear algebra, functional analysis, and group theory. They are particularly useful in solving equations and systems of equations, as well as proving mathematical theorems and properties.

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