MATRICES AND DETERMINATS )1 - Uuooidata.org

Remark 1.5. Gauss–Jordan elimination and LU decomposition arealgorithms which are used to determine whether a given matrix is invertibleand to find the inverse.RANK OF A MATRXLetA ∈M m.n (K) be a matrix with m rows and n colmns,⎛ a11⎜⎜ a21A = ⎜ ...⎜⎝am1aaa1222...m2............a1n ⎞⎟a2n⎟... ⎟⎟amn ⎠and k a natural number, such that 1 ≤ k ≤ min( m,n). If in A we choose k rowsi 1, i 2,..., i kand k columns j1, j2,..., jk, the elements which are found at the intersection ofthese rows and columns form a square matrix of order k:⎛ ai1 j1⎜⎜ai2j2⎜ ...⎜⎝ aikj1aaai j1 2iikj2 2...j2............aaai j1 ki2 kikj...jk⎞⎟⎟⎟ ∈⎟⎠M k (K)whose determinant is called the minor of order k for the matrix A.Let A ≠ , A ∈M m.n (K). It results that the matrix A has non-zero elements, then there areOm , nnonzero minors of an order k ≥ 1.Since the set of minors of matrix A is a finite set, there is anatural number r, 1 ≤ r ≤ min( m,n),such that there is at least one minor of order r differnt fromzero and all minors of orders bigger than r, if they exist, are zero.8

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