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7.1 SOLUTIONS
Notes: Students can profit by reviewing Section 5.3 (focusing on the Diagonalization Theorem) before working on this section. Theorems 1 and 2 and the calculations in Examples 2 and 3 are important for the sections that follow. Note that symmetric matrix means real symmetric matrix, because all matrices in the text have real entries, as mentioned at the beginning of this chapter. The exercises in this section have been constructed so that mastery of the Gram-Schmidt process is not needed.
Theorem 2 is easily proved for the 2 2 case:
If ,a b
Ac d
= then ( )2 21 ( ) 4 .2 a d a d b = + + If b = 0 there is nothing to prove. Otherwise, there are two distinct eigenvalues, so A must be diagonalizable.
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In each case, an eigenvector for is .db
1. Since 3 5
,5 7
TA A = = the matrix is symmetric.
2. Since 3 5
,5 3
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TA A
= the matrix is not symmetric.
3. Since 2 2
,4 4
TA A = the matrix is not symmetric.
4. Since 0 8 38 0 2 ,3 2 0
TA A
= =
the matrix is symmetric.
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5. Since 6 2 00 6 2 ,0 0 6
TA A
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=
the matrix is not symmetric.
6. Since A is not a square matrix TA A and the matrix is not symmetric.