Matrices for solving sets of linear equations

Matrices also appear in the solution of problems in linear algebra because they provide a compact way of discussing sets of equations. For example if we have three unknowns, Xu X2 and X3 which conform to the similtaneous equations  

This allows any algebraic manipulation to be carried out in the shorthand notation. For example, if we wish to find the values of X in Equation (A5.13), then we can see from Equation (A5.15) that, if we can find the inverse matrix of A, the solution will be 

To find the inverse matrix requires the introduction of the determinant. The determinant is related to the square matrices we use as representations for symmetry operations, but is a simple number formed in a systematic way from the elements of the matrix. To distinguish the determinant from a matrix it is written enclosed in straight lines, rather 

The evaluation of the determinant requires its simplification into sets of 2 x 2 subdeterminants, which can then be evaluated using the following prescription  

To evaluate a higher order determinant we choose a row of the determinant and form the product of each element with its co-factor. The co-factor is the sub-determinant formed by blocking the row and column for a given element and forming a determinant from the visible elements. For example, to form the co-factor Cn of the an element in Equation (A5.17)  

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