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Precalculus

Course: Precalculus>Unit 7

Lesson 13: Introduction to matrix inverses

Inverse matrices and matrix equations

In other videos, we've seen how matrices can represent systems of equations, and we've also seen how matrices whose determinant is zero don't have an inverse. In this video, we connect this two understanding, and by doing that way we understand how we can solve systems of equations with matrices. Created by Sal Khan.

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• How does the property of linear dependence relate to the question of whether there is a solution to a matrix system of equations?
• A set of vectors is linearly dependent if you can express one of the vectors as a sum of the others; in that case, you could remove that vector from your set and you wouldn't be able to express anything less. Some of the vectors carry redundant information.

Likewise, a system of equations is underconstrained if you can generate one of the equations by combining the others; that equation didn't add any new information, so you may as well have not had it.

So when you have a matrix, you can either interpret it as a representation of a system of equations, or you can interpret the rows as vectors. The row vectors are linearly independent exactly when the system of equations has no unique solution.
• That's a beautiful result. Is this how the concept of the determinant came about to begin with?
• how do u know A^-1 will have the same dimension as A?
• In general, f you have an axb matrix A and a cxd matrix B, the multiplication AB is not well-defined unless b=c.

A must be square to be invertible, so say A is an axa matrix. If we want the inverse of A, we know that A⁻¹ satisfies AA⁻¹=I, so the multiplication is well-defined. A⁻¹ must be ax(something).

We also know A⁻¹A is well-defined, so by the same logic, we know that A⁻¹ will be an axa matrix.