Scilab's inv() Function

In Scilab, the inv() function is a powerful tool designed to compute the inverse of a matrix. Consider the syntax:

inv(M)

In this context, M denotes a square matrix.

When executed, the inv() function yields the inverse matrix of M, provided it exists.

But what exactly is an inverse matrix? If we have a square matrix M, its inverse matrix, represented as M^-1, satisfies the condition where their product equates to the identity matrix of M's dimensions: $$ M \cdot M^{-1} = I $$

Let's delve into a practical example.

Suppose you define a 2x2 matrix and assign it to the variable A

A = [2, 1; 1, 3];

To determine its inverse, employ the inv() function and store the outcome in the variable B

B = inv(A);

To view the contents of B, simply use disp(B)

disp(B)

This will reveal A's inverse matrix.

0.6 -0.2
-0.2 0.4

For validation, compute the product of matrix A and its inverse B=inv(A)

A*B

The product is a 2x2 identity matrix, characterized by a primary diagonal of ones and all other elements being zero.

ans =
1. 0.
0. 1.

It's crucial to note that inverses don't exist for all matrices. Two primary conditions must be met:

1. The matrix is square, implying an equal number of rows and columns.
2. Its determinant isn't zero.

Should a matrix lack an inverse, executing the inv() function will trigger an error message.

So, why prioritize understanding the inverse matrix? Possessing this knowledge significantly streamlines various mathematical tasks, notably the resolution of linear equation systems.