Page 320 - Linear Models for the Prediction of Animal Breeding Values 3rd Edition

For instance, given that:
é 84ù 1 é 4 - 4ù é 0.50 - 0.50ù
0
1
A = ê ú then A = ê ú = ê ú
-
-
ë
ë 64 û (8)(4) (6)(4) - 6 8 û ë - 0.75 1.00 û
−1
−1
Note that A A = I = AA , as stated earlier. Calculating the inverse of a matrix
becomes more cumbersome as the order increases, and inverses are usually obtained
using computer programs. The methodology has not been covered in this text. It is
obvious from the above that an inverse of a non-diagonal matrix cannot be calculated
if the determinant is equal to zero. A square matrix with a determinant equal to zero
is said to be singular and does not have an inverse. A matrix with a non-zero deter-
minant is said to be non-singular.
−1
Note that (AB) = B A . The inverses of matrices may be required when solving
−1
−1
linear equations. Thus given the following linear equation:
Ab = y
−1
pre-multiplying both sides by A gives the vector of solutions b as:
b = A y
−1

A.3.6 Rank of a matrix

The rank of a matrix is the number of linearly independent rows or columns.
A square matrix with the rank equal to the number of rows or columns is said
to be of full rank. In some matrices, some of the rows or columns are linear
combinations of other rows or columns; therefore, the rank is less than the num-
ber of rows or columns. Such a matrix is not of full rank. Consider the following
set of equations:
3x + 2x + 1x = y
1 2 3 1
4x + 3x + 0x = y
1 2 3 2
7x + 5x + 1x = y
1 2 3 3
The third equation is the sum of the first and second equations; therefore, the vector
of solutions, x(x′ = [x x x ]), cannot be estimated due to the lack of information. In
1 2 3
other words, if the system of equations were expressed in matrix notation as:
é 32 1ù éx 1ù é y 1 ù
ê ú ê ú ê ú
ê
ê 43 0 ú ê 2 x ú = y 2ú
ê ë 75 1ú ê x 3û ú ë ê ys 3û ú
û ë
that is, as:

Dx = y
a unique inverse does not exist for D because of the dependency in the rows. Only
two rows are linearly independent in D and it is said of to be of rank 2, usually writ-
ten as r(D) = 2. When a square matrix is not of full rank, the determinant is zero and
hence a unique inverse does not exist.

304 Appendix A