Matrix Inverse
Overview
The matrix inverse for a given matrix is the matrix that when multiplied by the original matrix, returns the identity matrix. Not all matrices
have inverses.
{% A^{-1} A = I %}
Discussion
A matrix is singular if there ia linear combination (with at least one non-zero coefficient) of the columns of the matrix that equals the zero vector
{% \vec{0} %}.
{% A\vec{x} = \vec{0} %}
If {% A %} is an {% n \times n %} non-singular matrix, that means that there exist vectors {% \vec{u}_i %} such that
{% A \vec{u}_i = \vec{e}_i %}
Then, one can create a matrix {% U %}, such that the {% t^{th} %} column is equal to {% u_i %}. This implies
{% AU = I %}
That is, {% U %} is the inverse of {% A %}.
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