.
(a)
det(A)=−7, adj A=[−1−3−21],A−1=[173727−17];
(c)
det(A)=3, adj A=⎡⎣⎢−30651−821−5⎤⎦⎥,A−1=13adj A
.
(a) (57,87);
(b) (115,−45);
(c) (4,−2,2);
(d) (2,−1,2);
(e) (−23,23,13,0)
. −34
. (12,−34,1)T
.
(a) det(A)=0, so A is singular.
(b)
adj A=⎡⎣⎢−12−12−42−12−1⎤⎦⎥ andA adj A=⎡⎣⎢000000000⎤⎦⎥
. Hint: The solution of the linear system Ix = b is x=b.
.
(a) det(adj(A))=8 and det(A)=2;
(b) A=⎡⎣⎢⎢⎢⎢100004−610−12001−21⎤⎦⎥⎥⎥⎥
12. Hint: If det(A)=1, show that
and then apply the result from Exercise10 .
13. Hint: The (j, i) entry of QT is qij. Determine an expression for the (j, i) entry of Q−1 involving a cofactor of Q.
14. Do Your Homework.
15.
(d) Hint: Expand the determinant into cofactors along the third row.
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