We describe an algorithm to determinewhether a matrix group over a finite field, generatedby a given set of matrices, contains one of the classicalgroups or the special linear group.The algorithm is based onfindingelements withcertain propertiesbyindependent random selection from thegroup. Very few matrix groups other than the classical groups containsubsets of elements with the requiredproperties, and a classification of such matrix groups is the principaltheoretical basis of the algorithm. The classification, which is proved here, builds on work of Guralnick,Penttila, Saxl and the second author, and depends on the finite simple group classification.Moreover,probability estimates for finding such elements in classical groups are derived, and these are used to givea theoretical analysis ofthe performance of the algorithm. The algorithm has been implemented in the computer systems {\sf GAP} and{\scMagma}.

1991 Mathematics Subject Classification: primary 20G40, 11E57, 20-04,60B99; secondary 15A30.