Matrix Invertibility

What is Matrix Invertibility?

A matrix A is invertible if there exists another matrix B such that: AB = BA = I

where $I$ is the identity matrix. In this case, we write $B = A^{-1}$ and call it the inverse of $A$.

Key Properties

1. If A is invertible, its inverse is unique
2. (AB)⁻¹ = B⁻¹A⁻¹
3. (A⁻¹)⁻¹ = A
4. A matrix is invertible if and only if its determinant is non-zero

Connection to Linear Transformations

A matrix represents an invertible linear transformation if and only if:

• It preserves the dimension of the space
• It maps distinct vectors to distinct vectors
• It can be "undone" by another transformation

Connection to Systems of Equations

A matrix is invertible if and only if the system Ax = b has a unique solution for every b. This is why invertibility is crucial in solving systems of linear equations.