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013) that if A is regular so are all positive n powers of A. The inverse of A" is A~ . 02 to the case of arbitrary integral exponents. 02. 053). If k, / ^ 0 , then we put k= — n, 1= —m ( « , ra^O) and consider the product n A m A = B. This gives n A m = BA , and m E = BA A n m+n = BA whence -(m + n) A = β = -m-n A ~n =A ~m A If k > 0 and / < 0 we have k AA l k +l =A l A~ A l =A k +l and this means that we have proved the entire assertion. 14: Let A and Β be two regular square matrices of the same order.

06 that the determinant of an r-rowed minor o f U—for example, the one that consists of the first r rows—is non-zero. Likewise it follows (cf. 06a) that an r-rowed square minor, thus for example the one formed from the first r columns o f V, has a non-zero determinant. 013). In other words, A has at least one regular r-rowed square minor, whence p(A)^r. We shall now prove r is also an upper bound o f ρ (A). 071) where a and b are two arbitrary vectors. I f for example we take an arbitrary two-rowed square minor from the dyadic product a1b1 ü2bl (b»b ...

19: The ranks of the rows and the columns of a matrix are equal. 11 51 THE MINIMAL DECOMPOSITION OF A MATRIX This theorem gives rise to the following D E F I N I T I O N : The rank of the rows (or columns) of a matrix A is called the rank of the matrix and is written ρ (A). This definition means that the largest number of linearly independent vectors that can be chosen from the rows and columns is exactly ρ (A). 20 : Ifthe rank of a matrix A is equal to p(A) = p, there is at least one regular minor of order ρ and every square minor of order greater than ρ is singular.