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T = Find the inverse of Show that b1' bT, (a) (b) [2 4 [i S (c) [1 [3 31[2 4 2J[l s (d) [2 O1[2 [o 3j[l s Let a12 a = 31 [A11 A12 = [A2, A22 33 32 and B32 — [8,, 82, where the order of BJk is the same as AJ1. Starting with the condition aa' 13 PROBLEMS 41 determine the four matrix equations relating BJk and Aft (j, k result to find the inverse of 1—3!. 1, 2). Use this 124 212 121 Find the inverse of A lq L0 Note that A is (p x q). 1-32. Find the inverse of 0 D2 1—33. Use the results of Probs. 1—31 and 1—32 to find the inverse of b — [B11 B12 [o B22 where B11, B22 are square and nonsingular.
REFERENCES 33 In general, (1 —71) can be solved when r < ,n if the relations between the rows of a and c are identical. We define the augmented matrix, ci, for (1—71) as a11 a12 C1 = a2,, a,flfl afl,2 [a cJ (1—74) Cm When the rows of a and c are related in the same way, the rank of tz is equal to the rank of a. It follows that (1—71) has a solution only if the rank of the augmented matrix is equal to the rank of the coefficient matrix: = r(a) (1—75) Note that (1—75) is always satisfied when r(a) = m for arbitrary c.
11. RALSTON, A. and H. S. WILF: Mathematical Methods for Digital Computers, Vol. 2, Wiley, New York, 1967. 10. Wiley, 12. BEREZIN, I. S. and N. P. ZHIDKOV: Computing Methods, Vols. , 1965. 13. 14. 15. FORSYTHE, G. , and C. B. MALER: Computer Solution of Linear Algebraic Systems, Prentice-Hall, New York, 1967. VARGA, R. : Matrix Iterative Analysis, Prentice-Hall, New York, 1962. CONTE, S. : Elementary Numerical Analysis, McGraw-Hill, New York, 1965. PROBLEMS 1—1. 3 321 +713 510 056 (b) [2 1 6j[ 3 1 (c) —1 2 [1 21 [2 3[3 4j+2[[—i0 ii 3j4[l [i (d) —2152 [—3 4J 55 (e) [—i l][4 1 2 —3j [2 3 [ 1—2.