Test 1 Practice Questions (First set) (1) Consider the following linear ...
Test 1 Practice Questions (First set) (1) Consider the following linear ...
Test 1 Practice Questions (First set) (1) Consider the following linear ...
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(7) For<br />
A =<br />
⎡<br />
⎢<br />
⎣<br />
0 1 2<br />
0 2 4<br />
0 3 6<br />
1 4 8<br />
(a) Find a vector b in R 4 such that <strong>the</strong> system Ax = b is inconsistent;<br />
(b) Find a vector c in R 4 such that <strong>the</strong> system Ax = c is consistent.<br />
<br />
1 10<br />
(8) Let A =<br />
. Find a scalar λ such that A − λI2 fails to be invertible. For<br />
−3 12<br />
each scalar λ you find, find a nonzero vector x such that Ax = λx.<br />
(9) Answer <strong>the</strong> <strong>following</strong> questions<br />
(a) Describe <strong>the</strong> kernel of <strong>the</strong> matrix ⎣<br />
⎡<br />
⎤<br />
⎥<br />
⎦ ,<br />
1 1 1<br />
1 2 3<br />
1 3 5<br />
(b) Describe <strong>the</strong> image of <strong>the</strong> <strong>linear</strong> transformation T (x) = Ax for A = ⎣<br />
(10) For <strong>the</strong> matrix<br />
⎡<br />
A = ⎣<br />
0 1 0<br />
0 0 1<br />
0 0 0<br />
describe <strong>the</strong> images and kernels of <strong>the</strong> matrices A, A 2 , A 3 .<br />
⎤<br />
⎦;<br />
⎡<br />
1 1 1<br />
1 1 1<br />
1 1 1<br />
(11) <strong>Consider</strong> a n × p matrix A and a p × m matrix B such that ker(A) = {0} and<br />
ker(B) = {0}. Find ker(AB).<br />
(12) For two invertible n × n matrices A and B, determine which of <strong>the</strong> formulas below<br />
are necessarily true.<br />
(a) (A + B) 2 = A 2 + 2AB + B 2<br />
(b) A 2 is invertible and (A 2 ) −1 = (A −1 ) 2<br />
(c) A + B is invertible, and (A + B) −1 = A −1 + B −1<br />
(d) (A + B)(A − B) = A 2 − B 2<br />
(e) ABB −1 A −1 = In<br />
(f) ABA −1 = B<br />
(g) (ABA −1 ) 3 = AB 3 A −1<br />
(h) (In + A)(In + A −1 ) = 2In + A + A −1<br />
(i) A −1 B is invertible, and (A −1 B) −1 = B −1 A.<br />
2<br />
⎤<br />
⎦ ,<br />
⎤<br />
⎦.