We consider matrices of the following form:
G
n
(
a
1
,
a
2
,
⋯
,
a
n
−
1
,
b
1
,
b
2
⋯
b
n
)
=
{G_n}({a_1},{a_2}, \cdots ,{a_{n - 1}},{b_1},{b_2} \cdots {b_n}) =
(
β
i
,
j
)
,
1
≦
i
({\beta _{i,j}}),1 \leqq i
,
j
≦
n
j \leqq n
, where
a
1
,
⋯
,
a
n
−
1
,
b
1
,
⋯
,
b
n
{a_1}, \cdots ,{a_{n - 1}},{b_1}, \cdots ,{b_n}
are constants and
\[
β
i
,
j
=
b
j
,
j
≧
i
;
β
i
j
=
a
j
,
j
>
i
.
{\beta _i}_{,j} = {b_j},{\text { }}j \geqq i;{\text { }}{\beta _{ij}} = {a_j},{\text { }}j > i.
\]
We deduce in analytic form the determinant, inverse matrix, characteristic equation, and eigenvectors of
G
n
{G_n}
. Knowing these properties enables us to generate valuable test matrices by appropriately selecting the order and elements of
G
n
{G_n}
.