1.2.2 No solution; zero tight sides produce 3 lines through origin; right sides give and ; will give and ; any combination is solvable. 1.2.6 Both and give a line of solutions. All other give , . 1.2.8 The conditions for a straight line can be written as , . 1.2.11 Column 3 (column 2) - column 1. If then . 1.2.16 If then and give the point . If then and give the point . Halfway between is . 1.2.23 Four planes in 4-dimensional space normally meet at a point. The solution to is if has columns . The equations are , , , . 1.3.2 Subtract times equation 1 (or add times equation 1). The new second equation is . Then and . If the right side changes sign, so does the solution: . 1.3.4 Multiply by and subtract to find and . Pivots , . 1.3.7 is times . The new second pivot multiplying is or . Then . 1.3.10 The second pivot position will contain . If exchange with row 3. If (singular case) the second equation is . A solution is . 1.3.13 gives Subtract ×row 1 from row 2 and Subtract ×row 2 from row 3 Subtract × row 2 from row 3 1.3.15 (a) Another solution is . (b) If planes meet at two points, they meet along the whole line through those two points. (a) 1.3.17 (exchange 1 and 2, then 2 and 3) 1.3.19 , , (b) (rows 1 and 3 are not consistent) has infinitely many solutions. 1.3.26 (a) True (b) False because that entry can change (c) True. 1.3.28 . Change to would make the system singular (2 equal columns). 1.4.3 need multiplications; needs multiplications. 1.4.5 , , . With sides to and , the parallelogram goes to 1.4.7 Combinations of columns give , , . 1.4.11 True; False; True; False. , , , , . 1.4.15 1.4.22 Changing from to will change the third pivot from to . Changing from to will change the pivot from to no pivot. 1.4.27 = . , , 1.4.40 (a) True (b) False (c) True (d) False (take ). 1.4.42 By linearity agrees with . Also for all other columns of . 1.4.52 The . block si the Schur complement: blocks in 1.4.57 → 1.5.6 (a) Because by the time a pivot row is used, it is taken from , not ; (b) Row 3 of comes from multiplying row 3 of times the matrix . 1.5.7 ; after elimination . 1.5.8 In , the unknown is found in only one operation, requires two operations, and requires . The total is ⋯ ≈ . 1.5.11 Singular, no solution; Singular, infinitely many solutions; Nonsingular, one solution. 1.5.18 going downward gives ; upwards gives . 1.5.24 and : reverse steps to recover from : 1 times times times gives . 1.5.26 leads to zero in the second pivot position: exchange rows and the matrix will be OK. leads to zero in the third pivot position. In this case the matrix is singular. 1.5.31 gives . Then gives . Check that times is . 1.5.41 or its transpose has ; for the same has . 1.5.45 so three rotations for ; rotates around by . 1.6.1 If row 3 of were then gives , , . This has no solution. 1.6.4 ; ; . 1.6.6 (a) ⇒ . (b) . . 1.6.14 1.6.16 ; ; . 1.6.19 (1), (2), (5): triangular , symmetric , rational remain true for . 1.6.21 (a) In , equation 1 + equation 2 - equation 3 is . (b) The tight sides must satisfy . (c) Row 3 becomes a row of zeros-no third pivot. 1.6.24 From we get , an explicit inverse provided and are invertible. Second approach: If is not invertible, then for some nonzero . Therefore , or , and could not be invertible. (Note that is nonzero from .) 1.6.32 If has a column of zeros, so does . So is impossible. There is no . . 1.6.36 → → 1.6.41 (a) True ( has a row zeros) (b) False (matrix of all 1's) (c) (inverse of is ) (d) True ( is the transpose of ). 1.6.44 . The by superdiagonal. also has 's on the diagonal and 1.6.52 is not except when . In that case transpose to find: . 1.6.58 and are symmetric if and are symmetric. . 1.6.62 Total currents are Either way . 1.7.2 Det . by moving to the right side. 1.7.5 1.7.8 . 1.7.9 The pivots are , ; after a row exchange they are , .
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