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Tumbuhan.pdfQ:

Finding and Applying an Exponent in Mathematica

There are a lot of ways to compute the symbolic inverse of an matrix. For my project, I decided to use the Algebra package and its Inverse function. In its documentation, it states:

If A is an NxN matrix with real entries, the inverse is computed by
inverting Ax. x is the vector with components xi/Axi, where Xi
corresponds to the i-th row.

This is something I can understand. I then read a little bit further and found this statement:

If A is an NxN matrix with complex entries, or A is a NxN matrix
whose first column consists of only zeros and the entries of the second
column are real, then this command first computes the inverse of A
without the multiplications.

What I don’t understand is the “without the multiplications” part.
I have a matrix A, whose 1st column is all zeros (like the example) and I want to find out the inverse of this matrix. My question is:

How can I find the inverse of this matrix using the Inverse function?
Does the “without the multiplications” part mean that no multiplications are used? If this is not the case, can anyone show me a sample code?

Thanks in advance.

A:

The function inversion usually computes (a) the inverse of a matrix with real entries and (b) the inverse of an upper triangular matrix without the first row. This allows us to express the inverse of a matrix as a sum of products of the following structures:
(a) a(a’A)^{ -1}a’ (the first row of A, multiplied by) a'(a’A)^{ -1}a (all but the first column of A, multiplied by) a'(a’A)^{ -1}
(b) a’ (the first column of A, multiplied by) a'(a’A)^{ -1}a (all but the first row of A, multiplied by) a'(a’A)^{ -1}
Generally speaking, the first row of A does not have to be zero and neither does