By J J Connor

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Second, third 1—38. Determine the rank of (a) 1 3 7 5 2 4 —4 —10 3 (b) 2 3 —1 4 6 —2 —2 —3 1 1 2 —1 1—39. Let a be of order in x ii and rank r. Show that a has n — r columns which are linear combinations of r linear independent columns. Verify for 1234 a= 2 5 1 3 2 7 12 14 1—40. Using properties 3, 4, and 7 of determinants (see Sec. 1—7), deduce that the elementary operations do not change the rank of a matrix. For con- venience, consider the first r rows and columns to be linearly independent.

The defect for this system is 1. If a is of rank 1, the second row is a scalar multiple, say A, of the first row. Multiplying the second equation in (1—64) by 1/A, we have a12x2 + a13x3 = a11x1 + a12x2 + a13x3 = a11x1 + C1 c2/A (1—69) 2cr, the equations are inconsistent and no solution exists. Then, when a is of rank 1, (1—64) has a solution only if the rows of c are related in the same manner as the rows of a. If this condition is satisfied, the two equations in (1—69) are identical and one can be disregarded.

1—20, 1—21. f See Ref. 4, sect. 3 15, for a discussion of the general Laplace expansion method. The expansion in terms of cofactors for a iow Or a COlUmn is a special case of the general method. CRAMER'S RULE SEC. 1—9. The evaluation of a determinant, using the definition equation (1—44) or the cofactor expansion formula (1—46) is quite tedious, particularly when the array is large. A number of alternate and more efficient numerical procedures for evaluating determinants have been developed. These procedures are described in References 9—13.