Given a system of simultaneous equations in matrix form
Where is an invertable n x n square matrix,
is a variables column matrix containing
and
is a constants matrix containing
.
Then can be found with Cramer’s rule,
Cramer's rule
Where is formed by replacing the ith column of
with
.
Determinants are only defined for square matrices. This article states the formulae for a 2×2 matrix, a 3×3 matrix and the general case. The general case is used when finding determinants of 4×4 matrices or bigger and most often when solving simultaneous equations. These large matrices involve quite a deal of computation and in real world situations would usually be solved by computers.
Square matrix - The number of rows and number of columns are equal.
Diagonal matrix - All elements other than those in the main diagonal are 0.
Identity matrix - The identity matrix is a diagonal matrix where all elements of the main diagonal are 1.
Scalar matrix - A diagonal matrix with all diagonal elements alike.
Row matrix - The matrix consists of a single row of elements.
Column matrix - The matrix consiste of a single column of elements.
Symmetric matrix - A square matrix which is equal to its transpose.
Skew-symmetric matrix - A square matrix which is equal to the opposite of its transpose.
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