With some understanding of basic matrix manipulations, we're ready to begin using matrices to solve systems of linear equations. In this post, you'll learn a few standard tools for solving small systems - system defined by a small number of equations - by hand. Naturally, larger systems as found in fMRI will use computers to solve the equations, but you should understand what's going on when you push the buttons.
A11. Elementary row operations and elimination
This is just your standard algebraic manipulation to solve multiple simultaneous equations, e.g. dividing both sides of an equation by some constant to be able to simplify, but where the equations are represented as matrices:
A12. Cramer's Rule for solving small linear systems
According to Wikipedia:
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-sides of the equations.
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