Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
Linear Algebra Exercises-n-Answers.pdf
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172 <strong>Linear</strong> <strong>Algebra</strong>, by Hefferon<br />
∣ ∣ ∣ ∣∣∣<br />
Four.III.1.13 (a) (−1) 2+3 1 0<br />
∣∣∣ 0 2∣ = −2 (b) 1 2<br />
∣∣∣ (−1)3+2 −1 3∣ = −5 (c) −1 1<br />
(−1)4 0 2∣ = −2<br />
Four.III.1.14 (a) 3 · (+1)<br />
∣ 2 2<br />
∣ ∣ ∣∣∣ 3 0∣ + 0 · (−1) 1 2<br />
∣∣∣ −1 0∣ + 1 · (+1) 1 2<br />
−1 3∣ = −13<br />
(b) 1 · (−1)<br />
∣ 0 1<br />
∣ ∣ ∣∣∣ 3 0∣ + 2 · (+1) 3 1<br />
∣∣∣ −1 0∣ + 2 · (−1) 3 0<br />
−1 3∣ = −13<br />
(c) 1 · (+1)<br />
∣ 1 2<br />
∣ ∣ ∣∣∣ −1 3∣ + 2 · (−1) 3 0<br />
∣∣∣ −1 3∣ + 0 · (+1) 3 0<br />
1 2∣ = −13<br />
⎛<br />
⎛<br />
Four.III.1.15 adj(T ) = ⎝ T ⎞<br />
+<br />
∣ 5 6<br />
∣ ∣ ⎞<br />
∣∣∣ 8 9∣ − 2 3<br />
∣∣∣ 8 9∣ + 2 3<br />
5 6∣<br />
1,1 T 2,1 T 3,1<br />
T 1,2 T 2,2 T 3,2<br />
⎠ =<br />
−<br />
∣ 4 6<br />
∣ ∣ ⎛<br />
⎞<br />
∣∣∣ 7 9∣ + 1 3<br />
∣∣∣ 7 9∣ − 1 3<br />
−3 6 −3<br />
4 6∣<br />
= ⎝ 6 −12 6 ⎠<br />
T 1,3 T 2,3 T 3,3 ⎜<br />
⎝<br />
+<br />
∣ 4 5<br />
∣ ∣ ⎟ −3 6 −3<br />
∣∣∣ 7 8∣ − 1 2<br />
∣∣∣ 7 8∣ + 1 2<br />
⎠<br />
4 5∣<br />
⎛<br />
⎛<br />
Four.III.1.16 (a) ⎝ T ⎞<br />
∣ 0 2<br />
0 1∣<br />
−<br />
∣ 1 4<br />
0 1∣<br />
∣ 1 4<br />
⎞<br />
0 2∣<br />
1,1 T 2,1 T 3,1<br />
T 1,2 T 2,2 T 3,2<br />
⎠ =<br />
−<br />
∣ −1 2<br />
1 1∣<br />
∣ 2 4<br />
1 1∣<br />
−<br />
∣ 2 4<br />
⎛<br />
−1 2∣<br />
= ⎝ 0 −1 2<br />
⎞<br />
3 −2 −8⎠<br />
T 1,3 T 2,3 T 3,3 ⎜<br />
⎝<br />
∣ −1 0<br />
1 0∣<br />
−<br />
∣ 2 1<br />
1 0∣<br />
∣ 2 1<br />
⎟ 0 1 1<br />
⎠<br />
−1 0∣<br />
( ) ( ∣ T1,1 T<br />
(b) The minors are 1×1:<br />
2,1<br />
4 ∣ − ∣ ∣−1 ∣ )<br />
( )<br />
4 1<br />
=<br />
T 1,2 T 2,2 − ∣ ∣2 ∣ ∣<br />
∣3 ∣ = .<br />
−2 3<br />
( )<br />
0 −1<br />
(c)<br />
−5 1<br />
⎛<br />
⎛<br />
(d) ⎝ T ⎞<br />
∣ 0 3<br />
8 9∣<br />
−<br />
∣ 4 3<br />
8 9∣<br />
∣ 4 3<br />
⎞<br />
0 3∣<br />
1,1 T 2,1 T 3,1<br />
T 1,2 T 2,2 T 3,2<br />
⎠ =<br />
−<br />
∣ −1 3<br />
1 9∣<br />
∣ 1 3<br />
1 9∣<br />
−<br />
∣ 1 3<br />
⎛<br />
⎞<br />
−24 −12 12<br />
−1 3∣<br />
= ⎝ 12 6 −6⎠<br />
T 1,3 T 2,3 T 3,3 ⎜<br />
⎝<br />
∣ −1 0<br />
1 8∣<br />
−<br />
∣ 1 4<br />
1 8∣<br />
∣ 1 4<br />
⎟ −8 −4 4<br />
⎠<br />
−1 0∣<br />
⎛<br />
Four.III.1.17 (a) (1/3) · ⎝ 0 −1 2<br />
⎞ ⎛<br />
⎞<br />
0 −1/3 2/3<br />
3 −2 −8⎠ = ⎝1 −2/3 −8/3⎠<br />
( ) (<br />
0 1 1<br />
)<br />
0 1/3 1/3<br />
4 1 2/7 1/14<br />
(b) (1/14) · =<br />
−2 3 −1/7 3/14<br />
( ) ( )<br />
0 −1 0 1/5<br />
(c) (1/ − 5) ·<br />
=<br />
−5 1 1 −1/5<br />
(d) The matrix has a zero determinant, and so has no inverse.<br />
⎛<br />
⎞ ⎛<br />
⎞<br />
T 1,1 T 2,1 T 3,1 T 4,1 4 −3 2 −1<br />
Four.III.1.18 ⎜T 1,2 T 2,2 T 3,2 T 4,2<br />
⎟<br />
⎝T 1,3 T 2,3 T 3,3 T 4,3<br />
⎠ = ⎜−3 6 −4 2<br />
⎟<br />
⎝ 2 −4 6 −3⎠<br />
T 1,4 T 2,4 T 3,4 T 4,4 −1 2 −3 4<br />
Four.III.1.19 The determinant ∣ ∣∣∣ a b<br />
c d∣<br />
expanded on the first row gives a · (+1)|d| + b · (−1)|c| = ad − bc (note the two 1×1 minors).<br />
Four.III.1.20 The determinant of ⎛<br />
⎝ a b c<br />
⎞<br />
d e f⎠<br />
g h i<br />
is this.<br />
a ·<br />
∣ e f<br />
∣ ∣ ∣∣∣ h i ∣ − b · d f<br />
∣∣∣ g i ∣ + c · d e<br />
g h∣ = a(ei − fh) − b(di − fg) + c(dh − eg)
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