CONTINUUM MECHANICS for ENGINEERS

Finally, we note that if the det A = 0, the matrix is said to be singular. It may
be easily shown that every 3 × 3 skew-symmetric matrix is singular. Also,
the determinant of the diagonal matrix, D, is simply the product of its
diagonal elements: det D = D 11D 22… D NN.
Example 2.4-3
Show that for matrices A and B, det AB = det BA = det A det B.
Solution
Let C = AB, then C ij = A ikB kj and from Eq. (2.4-11)
but from Eq 2.4-12,
so now
det C
= A iqB q1A jmB m2A knB n3
εijk = ε A ijk iqA jmA knB q1B m2B n3
ε AiqAjmAkn = det A
ijk
εqmn det C = det AB = ε Bq1Bm2Bn3 det A = det B det A
qmn
By a direct interchange of A and B, det AB = det BA.
Example 2.4-4
Use Eq 2.4-9 and Eq 2.4-10 to show that det A = det A T .
Solution
Since
= εijkCC i1 j2Ck3 cofactor expansion by the first column here yields
A
T =
A A A
A A A
A A A
11 21 31
12 22 32
13 23 33
A T A22 A32
A21 A31
= A11 − A12 + A
A A A A
23 33
23 33
13
A A
A A
21 31
22 32