MATRICES AND DETERMINATS )1 - Uuooidata.org

Definition 1.6. Let A ∈ M m.n (K) a non-zero matrix. The matrix A has rank r, rankA=r,if r is the greatest order of any non-zero minor in the matrix (the order of a minor being the sizeof the square sub-matrix of which it is the determinant). Obviouslly, all minors of order greaterthan r, if exist, are zero.If A is zero matrix, then rank( O ,)=0.m nTheorem 1.7. Let A ≠ O m , n, a matrix. The natural number r is the rank of the matrix A if andonly if there is in A a non-zero minor of order r and all minors of r+1 order, if exist, are zero.ExampleCompute the rank for the matrix:⎛32 − 5 4 ⎞⎜⎟A = ⎜3−13 − 3⎟.⎜⎟⎝35 −1311 ⎠Proof.We found a minor of order 2, for example:332= −9≠ 0−1We compute all minors of order three, and will find that all are zero:3332−15− 53−13=3332−154− 311=333− 53−134− 311=2−15− 53−134− 311= 0 . Therefore rank A=2.Exercices.1) Compute the determinants:− 2 3 5a) 11 − 6 8 ;7 13 −19