In mathematics the determinant is an operator which has certain useful identities.

det
(
I
n
)
=
1
where *I*_{n} is the *n* × *n* identity matrix.

det
(
A
T
)
=
det
(
A
)
.

det
(
A
−
1
)
=
1
det
(
A
)
=
det
(
A
)
−
1
.

For square matrices *A* and *B* of equal size,

det
(
c
A
)
=
c
n
det
(
A
)
for an *n* × *n* matrix.

If *A* is a triangular matrix, i.e. *a*_{i,j} = 0 whenever *i* > *j* or, alternatively, whenever *i* < *j*, then its determinant equals the product of the diagonal entries:
det
(
A
)
=
a
1
,
1
a
2
,
2
⋯
a
n
,
n
=
∏
i
=
1
n
a
i
,
i
.

The following identity holds for a Schur complement of a square matrix:

The Schur complement arises as the result of performing a block Gaussian elimination by multiplying the matrix *M* from the right with the "block lower triangular" matrix

L
=
[
I
p
0
−
D
−
1
C
I
q
]
.
Here *I*_{p} denotes a *p*×*p* identity matrix. After multiplication with the matrix *L* the Schur complement appears in the upper *p*×*p* block. The product matrix is

M
L
=
[
A
B
C
D
]
[
I
p
0
−
D
−
1
C
I
q
]
=
[
A
−
B
D
−
1
C
B
0
D
]
=
[
I
p
B
D
−
1
0
I
q
]
[
A
−
B
D
−
1
C
0
0
D
]
.
That is, we have shown that

[
A
B
C
D
]
=
[
I
p
B
D
−
1
0
I
q
]
[
A
−
B
D
−
1
C
0
0
D
]
[
I
p
0
D
−
1
C
I
q
]
,