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Unformatted text preview: Solutions to practice problems for Test 4
1. Label the following statements as true or false.
false (a) For any scalar k and square matrix A, det(kA) = k det(A).
true (b) For any n × n matrices, A and B , (det A)(det B ) = det(AB ).
true (c) For any n × n matrices, A and B , det(AB ) = det(BA).
false (e) The dimension of the eigenspace corresponding to an eigenvalue
λ is equal to the multiplicity of λ as an eigenvalue.
true (d) A square matrix A is not invertible if and only if 0 is an eigenvalue
of A.
false (f) If A is diagonalizable, then A is invertible.
false (g) If A is invertible, then A is diagonalizable.
false (h) For any scalar c, cv = c v .
2 =u 2 +v 2 false (i) u+v for all u and v . true (j) u − v = u + v if and only if u and v are orthogonal. true (k) For any matrix, A, a vector v is in Nul A if and only if w • v = 0
for all vectors w in Row A.
false (l) If X is any orthogonal basis of Rn , then KX (w ) = w for all
vectors w in Rn . ...
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This document was uploaded on 06/18/2011.
 Spring '09
 Matrices, Scalar

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