A simple condition for when the matrix inverse exists | Linear algebra makes sense

TL;DR

Matrix inverses can be understood by examining left inverses, which undo the original matrix's action. A left inverse exists if the matrix doesn't lose information or map different vectors to the same vector. Non-square matrices never have inverses, while for square matrices, the left inverse is equal to the inverse.

Transcript

In a linear algebra course, Matrix inverses are something that get a lot of attention. There are all these formulas thrown at you like for the determinants, the inverses of a 2x2 matrices, cramner’s rule etc etc. These formulas are great, but I think they can obscure the very simple idea behind what an inverse actually is, and when it exists. You’v... Read More

Key Insights

• 🍃 Matrix inverses can be understood by examining left inverses, which undo the original matrix's action.
• 😚 A left inverse exists if the matrix doesn't lose information or map different vectors to the same vector.
• 😚 Non-square matrices never have inverses, as they always lose information.
• 🗨️ For square matrices, the left inverse is equal to the inverse, simplifying the process of finding an inverse.
• 😚 Checking if a matrix loses information can be done by examining how many vectors get mapped to zero.
• 😚 The existence of an inverse depends on whether the matrix loses information or not.
• 🦻 Understanding matrix inverses intuitively can be aided by solving examples and finding patterns.

Questions & Answers

Q: What is the difference between a left inverse and an inverse?

A left inverse undoes the action of the original matrix, while the inverse undoes both the original matrix and its left inverse. They are not always the same thing.

Q: Can a matrix have a left inverse but not an inverse?

Yes, if the matrix loses information or maps different vectors to the same vector, it will have a left inverse but not an inverse.

Q: Do all square matrices have inverses?

No, not all square matrices have inverses. It depends on whether the matrix loses information or maps different vectors to zero.

Q: How can you check if a matrix loses information?

By examining how many different vectors get mapped to zero. If there is more than one vector, the matrix loses information.

Summary & Key Takeaways

• Matrix inverses are often taught with complex formulas, but understanding the concept behind inverses is crucial.

• A left inverse is a matrix that undoes the original matrix's action, effectively doing nothing.

• The existence of a left inverse depends on whether the matrix loses information or maps different vectors to the same vector.