structural:reformatting
Linear equations

Matrices can be used to compactly write and work with multiple linear equations, that is, systems of linear equations. For example, if A is an m -by- n matrix, x designates a column vector (that is, n ×1-matrix) of n variables x, x,..., x, and b is an m ×1-column vector, then the matrix equation

is equivalent to the system of linear equations

Using matrices, this can be solved more compactly than would be possible by writing out all the equations separately. If n = m and the equations are independent, then this can be done by writing

where A is the inverse matrix of A. If A has no inverse, solutions—if any—can be found using its generalized inverse.