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A number of inversion procedures based on (1—51) have been developed. These methods are described in Ref. 9—13. 1—11. ELEMENTARY OPERATIONS ON A MATRIX The elementary operations on a matrix are:' 1. 2. 3. The interchange of two rows or of two columns. The multiplication of the elements of a row or a column by a number other than zero. The addition, to the elements of a row or column, of k times the corresponding element of another row or column. These operations can be effected by premultiplying (for row operation) or postmultiplying (for column operation) the matrix by an appropriate matrix, called an elementary operation matrix.

A1Br1, •. rb] A2Brb AraB2 Tb. What canyou conclude about Suppose ra (e) (f) A1 is orthogonal toB1,B2 Br? Utilize these results to find the rank of Suppose ra = = rb —1/2 1/2 0 1 0 —1/2 1/2 1 1 0 1 —1 1 1 1 1 2 Show that r(e) = s. s. 1 Verify for U 1—45. Consider the m x n system a12 021 022 Gm i 2 X1 C1 0 C2 X,, 0R111 Let = . j . 1, 2,. . , fl2 Using (b), we write (a) as in = 1, 2 (c) Now, suppose a is of rank r and the first r rows are linearly independent. Then, A3x = A1 (a) j k = r + 1, r + 2 = m Show that the system is consistent only if k= Ck i• + 1, r + 2,..

If each element in one row (or one column) is expressed as the sum of two terms, then the determinant is equal to the sum of two determinants, in each of which one of the two terms is deleted in each element of that row (or column). If to the elements of any row (column) are added k times the corresponding elements of any other row (column), the determinant is unchanged. We demonstrate these properties for the case of a second-order matrix. Let a = [a31 [a21 a22 The determinant is a! = a11a22 — a12a21 Properties 1 and 2 are obvious.

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