Sm Appendices Niku

I ntrod ntroduc ucti tion on to to Rob Robotics otics Analysis, Control, Applications

Solution Manual

Saeed B. Niku

©Copyri opyrighted 2010.  This  This solut lution ion manual may not be copied ied, posted, made availab ilable to students, plac laced on BlackBoard ackBoard or any other ele electronic ctronic media dia system and and the Inte Internet without without prior prior expresse expressed d consent of the copyright owner.

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APPENDICES Problem A.1 Show that the determinant of a matrix can be calculated by picking any row or column. Estimated student timeto complete: 10-15 minutes Prerequisite knowledge required: Text Section(s) A.1 Solution: For an arbitrary matrix:

⎡a b c⎤ A = ⎢ d e f ⎥  ⎢ ⎥ ⎢⎣ g h i ⎥⎦ det A = a( ei − fh) − b( di − fg) + c( dh− eg)

= aei − afh − bdi + bfg + cdh− ceg det A = −d ( bi − ch) + e( ai − cg) − f ( ah− bg)

= −dbi + dch+ eai − ecg − fah+ fbg det A = −b( di − fg) + e( ai − cg) − h( af − cd)

= −bdi + bfg+ eai − ecg − haf + hcd  They are all the same.

©Copyrighted 2010.  This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner.

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Problem A.2 Calculate the determinant of the following (4 × 4) matrix.

⎡1 ⎢0 A= ⎢ ⎢3 ⎢ ⎣1

1 0 0⎤ 1 2 0⎥ ⎥ 0 1 1⎥ ⎥ 0 0 1⎦

Estimated student timeto complete: 5 minutes Prerequisite knowledge required: Text Section(s) A.1. Solution: det A = 1(1(1− 0) − 0 + 0) −1(0 − 2(3− 1) − 0) + 0 − 0 = 1+ 4 = 5

©Copyrighted 2010.  This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner.

4 Problem A.3 Calculate the inverse of the following matrix using method 1:

⎡1 1 2⎤ B = ⎢0 1 0⎥ ⎢ ⎥ ⎢⎣2 0 3⎥⎦ Estimated student timeto complete: 5-10 minutes Prerequisite knowledge required: Text Section(s) A.1. Solution: det B = 1(3− 0) − 0 + 2(−2) = 3 − 4 = −1

⎡1 0 B T = ⎢1 1 ⎢ ⎢⎣ 2 0 ⎡3 adjB = ⎢ 0 ⎢ ⎢⎣ −2

2⎤ 0⎥ ⎥ 3⎥⎦ −3 −2⎤ −1 0 ⎥⎥ 2 1 ⎥⎦ ⎡ −3 3 2 ⎤ B−1 = ⎢ 0 1 0 ⎥ ⎢ ⎥ ⎢⎣ 2 −2 −1⎥⎦

©Copyrighted 2010.  This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner.

5 Problem A.4 Calculate the inverse of the following matrix using method 2:

⎡1 0 1⎤ C = ⎢0 2 1⎥ ⎢ ⎥ ⎢⎣ 3 1 0⎥⎦ Estimated student time to complete: 10-15 minutes Prerequisite knowledge required: Text Section(s) A.1. Solution:

⎡1 0 1⎤ ⎡ x11 x12 x13 ⎤ ⎡1 0 ⎢ 0 2 1⎥ ⎢ x x x ⎥ = ⎢0 1 ⎢ ⎥ ⎢ 21 22 23 ⎥ ⎢ ⎢⎣ 3 1 0⎥⎦ ⎢⎣ x31 x32 x33 ⎥⎦ ⎢⎣0 0 ⎧ x11 + x31 = 1 ⎧ x11 = 1/ 7 ⎪ ⎪ → ⎨ x31 = 6/ 7 ⎨2x21 + x31 = 0 ⎪3x + x = 0 ⎪ ⎩ x21 = −3/ 7 ⎩ 11 21 ⎧ x12 + x32 = 0 ⎧ x12 = −1/7 ⎪ ⎪ → ⎨ x22 = 3/ 7 ⎨2x22 + x32 = 1 ⎪3x + x = 0 ⎪ ⎩ x32 = 1/ 7 ⎩ 12 22 ⎧ x31 + x33 = 0 ⎧ x13 = 2/ 7 ⎪ ⎪ → ⎨ x23 = 1/ 7 ⎨2x23 + x33 = 0 ⎪3x + x = 1 ⎪ x = −2/ 7 ⎩ 13 23 ⎩ 33

0⎤ 0⎥ ⎥ 1⎥⎦

1 2⎤ ⎡1 ⎢ 7 −7 7 ⎥ ⎢ ⎥ 3 3 1 − −1 ⎥ C =⎢ ⎢7 7 7⎥ ⎢6 ⎥ 1 −2⎥ ⎢ ⎢⎣ 7 7 7 ⎥⎦

©Copyrighted 2010.  This solution manual may not be copied, posted, made available to students, placed on BlackBoard or any other electronic media system or the Internet without prior expressed consent of the copyright owner.