Solution
The definition of a determinant gives the relation y=y(t) =
∑
(−1)mai 1 (t)aj 2 (t)where the summation is over all permutations of the integers (1 ,2). Differentiating
this relation gives
dy
dt =d
dt(∑
(−1)mai 1 (t)aj 2 (t))
=∑
(−1)m[
da i 1 (t)
dt aj^2 (t) + ai^1 (t)daj 2 (t)
dt]which has the expanded form
dy
dt=∣∣
∣∣
∣da 11 (t)
dtda 12 (t)
dt
a 21 (t) a 22 (t)∣∣
∣∣
∣+∣∣
∣∣
∣a 11 (t) a 12 (t)
da 21 (t)
dtda 22 (t)
dt∣∣
∣∣
∣Minors and Cofactors
Associated with each element apq of a n×nsquare matrix Aare the quantities
mpq and cpq called the minor and cofactor of the element apq. The minor mpq of an
element apq is the determinant of the (n−1)×(n−1) matrix formed by deleting the row
and column of Awhich contains the element apq.The cofactor of apq is then defined
as cpq = (−1)p+qmpq.That is, the cofactor is the minor with the appropriate plus or
minus sign (−1)p+qwhich is determined by the row number pand column number q
of the element apq.The matrix containing the cofactor elements cpq of apq is written
C= (cpq)n×nand is called the cofactor matrix associated with A. The cofactor matrix
has the property that AC T=diag [|A|,|A|,... ,|A|] = |A|I, where |A|is the determinant
of A.
Example 10-18. ((Minors and Cofactors) For the matrix A=
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
calculate the cofactor matrix C= (cij) 3 × 3 and then calculate AC T.
Solution The minor of element aij is obtained by crossing out the row and column
containing aij and then taking the determinant of the remaining elements. One finds
m 11 =∣∣
∣∣a^22 a^23
a 32 a 33∣∣
∣∣,m 12 =∣∣
∣∣a^21 a^23
a 31 a 33∣∣
∣∣,m 13 =∣∣
∣∣a^21 a^22
a 31 a 32∣∣
∣∣m 21 =∣∣
∣∣a^12 a^13
a 32 a 33∣∣
∣∣,m 22 =∣∣
∣∣a^11 a^13
a 31 a 33∣∣
∣∣,m 23 =∣∣
∣∣a^11 a^12
a 31 a 32∣∣
∣∣m 31 =∣∣
∣∣a^12 a^13
a 22 a 23∣∣
∣∣,m 32 =∣∣
∣∣a^11 a^13
a 21 a 23∣∣
∣∣,m 33 =∣∣
∣∣a^11 a^12
a 21 a 22∣∣
∣∣c 11 = m 11 ,c 12 =−m 12 ,c 13 = m 13
c 21 =−m 21 ,c 22 = m 22 ,c 23 =−m 23
c 31 = m 31 ,c 32 =−m 32 ,c 33 = m 33AC T=
a 11 a 12 a 13
a 21 a 22 a 23
a 31 a 32 a 33
c 11 a 21 a 31
a 12 a 22 a 32
a 13 a 23 a 33
=
|A| 0 0
0 |A| 0
0 0 |A|