HP HP-15C - Part III HP-15 C Advanced Functions

Section 4: Using Matrix Operations 117
Singular and Nearly Singular Matrices
A matrix is singular if and only if its determinant is zero. The
determinant of a matrix is equal to (-l)r times the product of the
diagonal elements of U, where U is the upper-diagonal matrix of
the matrix's LU decomposition and r is the number of row
interchanges in the decomposition. Then, theoretically, a matrix is
singular if at least one of the diagonal elements of U, the pivots, is
zero; otherwise it is nonsingular.
However, because the HP-15C performs calculations with only a
finite number of digits, some singular and nearly singular matrices
can't be distinguished in this way. For example, consider the
matrix
B =
1 0 3 3
0 0
-LU,
which is singular. Using 10-digit accuracy, this matrix is
decomposed as
LU =
.3333333333 1
3 3
0 10-10
which is nonsingular. The singular matrix B can't be distin-
guished from the nonsingular matrix
D =
3 3
.9999999999 1
since they both have identical calculated L U decompositions.
On the other hand, the matrix
A =
3 3 "I
1 .9999999999
0 -1CT10
= LU