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Math 105
Semester Review - Solutions
T
T
2
h) ( ) ( ) ( ) ( ) ( )
det
i) det adj( )
AA = det A det A = det A det A = 27 = 729
( ) ⎡ ( A)
T
( ⎣ ⎤⎦
) ( A)
A = det cof = det cof
8 −1 3
= 11 2 −6
−13 5 12
( )
2 −6 11 −6 11 2
= 8 + + 3
5 12 −13 12 −13 5
= 8⋅ 54 + 54 + 3⋅ 81 = 729
T
2. A square matrix A is called skew-symmetric if A = − A.
a) Prove that if A is invertible and skew-symmetric, then
−
To prove: ( A )
−1
T
LS = ( A )
T
−1
= ( A )
( )
−1
1
T
−1
−1
=−A
1
A −
T
= − A since A is skew-symmetric ( A =−A
)
=−A
is skew-symmetric.
= RS
T
b) Prove that A , A + B and kA are skew-symmetric if A and B are skew symmetric.
T
To prove: ( ) T T
A =−A
LS =
T
( A )
T
T
( A) since A is skew-symmetric
= −
=−A
= RS
T
To prove: ( A + B) T
= − ( A+B
)
( )
LS = A + B
T
= +
A
B
T
T
=−A−B since A and B are skew-symmetric
= RS
( A B)
=− +
Winter 2006 Martin Huard 2