Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

420 VECTORS AND MATRICES [APP. A

Fig. A-5

The final Steprin Algorithm A-2 in Fig. A-6 changes the first pivot to 1.

Fig. A-6

Algorithms A-1 and A-2 show that any matrix is row equivalent to at least one matrix in row canonical form.
Actually, one proves in linear algebra that such a matrix is unique; it is called therow canonical formofA.

Theorem A.5:Any matrixAis row equivalent to a unique matrix in row canonical form.

Matrix Solution of a System of Linear Equations

Consider a systemSof linear equations or, equivalently, a matrix equationAX=Bwith augmented matrix
M=[A, B]. The system is solved by applying the above Gaussian elimination algorithm toMas follows.