It is sometimes convenient to think of the determinant of a matrix A as a function of its columns written
as vectors v 1 , v 2 ,…, v n . We write det(A) = det(v 1 , v 2 ,…,
v n ). Then, by the above observations, we have:
det(v 1 ,…, 0,…, v n ) = 0
det(v 1 ,…, v i ,…, v j ,…, v n ) = –det(v 1 ,…, v j ,…, v i ,…, v n )
det(v 1 ,…, v,…, v,…, v n ) = 0
We also have:
4. det(v 1 ,…, v + w,…, v n ) = det (v 1 ,…, v,…, v n )
+ det(v 1 ,…, w,…, v n )
and
5. det(v 1 ,…, kv,…, v n ) = k det(v 1 ,…, v,…, v n )
and consequently
6. The value of det(A) is not altered if a multiple of one column is added to another column: det(v 1 ,…, v i + kv j ,…, v n ) = det(v 1 ,…,
v i ,…, v n )
These results follow from the definition of the determinant. The corresponding results about rows are
also valid.
CRAMER’S RULE shows that the notion of a determinant is precisely the concept needed to solve simultaneously linear equations. We have:
A system of simultaneous linear equations has a
(unique) solution if the determinant of the corresponding matrix of coefficients is not zero.
Cramer’s rule goes further and provides a formula for
the solution of a system in terms of determinants.
The determinant has another important property.
After some algebraic work it is possible to show:
The determinant of the product of two n × n
matrices A and B is the product of their
determinants:
det(AB) = det(A) × det(B)
The determinant of the IDENTITY MATRIX I is one. If a
square matrix A is invertible, then the equation:
1 = det(I) = det(A · A
–1 ) = det(A) · det(A
–1 )
shows that det(A) is not zero and that
We have:
If a matrix is invertible, then its determinant is
not zero.
The converse is also true:
If the determinant of a matrix is not zero, then
the matrix is invertible.
To see why this holds, suppose that A is a matrix
with nonzero determinant. Let e i denote the ith column
of the identity matrix. By Cramer’s rule, since the determinant is not zero, the system of equations Ax = e i has a
solution x = s i , say. Set B to be the matrix with ith column s i . Then AB = I. This shows at least that A has a
“right inverse” B. To complete the proof, let A
T denote
the transpose of A, that is, the matrix obtained from A
by interchanging its rows and columns. Since the determinant can be viewed equivalently well as a function of
the rows of the matrix as its columns, we have that
det(A
T ) = det(A). Since the determinant of A
T is also
nonzero, there is a matrix C so that A
T C = I. One can
check that the transpose of the product of two matrices is
the reverse product of their transposes. We thus have:
C
T
A = (A
T
C)
T = I
T = I. This shows that the matrix A also
has a left inverse C
T . The left and right inverses must be
equal, since C
T = C
T I = C
T AB = IB = B. Thus the matrix
B is indeed the full inverse matrix to A: AB = BA = I.
See also INVERSE MATRIX.
diagonal Any line joining two nonadjacent vertices
of a POLYGON is called a diagonal of the polygon. For
example, a square has two diagonals, each cutting the
figure into two congruent right-angled triangles, and a
pentagon has five different diagonals. There are no
diagonals in a triangle. In general, a regular ngon has
distinct diagonals.
A diagonal for a POLYHEDRON is any line joining
two vertices that are not in the same face. A cube, for
n(n – 3)
———–
2
1
det(A
–1 ) = ———
det(A)
128 diagonal
as vectors v 1 , v 2 ,…, v n . We write det(A) = det(v 1 , v 2 ,…,
v n ). Then, by the above observations, we have:
det(v 1 ,…, 0,…, v n ) = 0
det(v 1 ,…, v i ,…, v j ,…, v n ) = –det(v 1 ,…, v j ,…, v i ,…, v n )
det(v 1 ,…, v,…, v,…, v n ) = 0
We also have:
4. det(v 1 ,…, v + w,…, v n ) = det (v 1 ,…, v,…, v n )
+ det(v 1 ,…, w,…, v n )
and
5. det(v 1 ,…, kv,…, v n ) = k det(v 1 ,…, v,…, v n )
and consequently
6. The value of det(A) is not altered if a multiple of one column is added to another column: det(v 1 ,…, v i + kv j ,…, v n ) = det(v 1 ,…,
v i ,…, v n )
These results follow from the definition of the determinant. The corresponding results about rows are
also valid.
CRAMER’S RULE shows that the notion of a determinant is precisely the concept needed to solve simultaneously linear equations. We have:
A system of simultaneous linear equations has a
(unique) solution if the determinant of the corresponding matrix of coefficients is not zero.
Cramer’s rule goes further and provides a formula for
the solution of a system in terms of determinants.
The determinant has another important property.
After some algebraic work it is possible to show:
The determinant of the product of two n × n
matrices A and B is the product of their
determinants:
det(AB) = det(A) × det(B)
The determinant of the IDENTITY MATRIX I is one. If a
square matrix A is invertible, then the equation:
1 = det(I) = det(A · A
–1 ) = det(A) · det(A
–1 )
shows that det(A) is not zero and that
We have:
If a matrix is invertible, then its determinant is
not zero.
The converse is also true:
If the determinant of a matrix is not zero, then
the matrix is invertible.
To see why this holds, suppose that A is a matrix
with nonzero determinant. Let e i denote the ith column
of the identity matrix. By Cramer’s rule, since the determinant is not zero, the system of equations Ax = e i has a
solution x = s i , say. Set B to be the matrix with ith column s i . Then AB = I. This shows at least that A has a
“right inverse” B. To complete the proof, let A
T denote
the transpose of A, that is, the matrix obtained from A
by interchanging its rows and columns. Since the determinant can be viewed equivalently well as a function of
the rows of the matrix as its columns, we have that
det(A
T ) = det(A). Since the determinant of A
T is also
nonzero, there is a matrix C so that A
T C = I. One can
check that the transpose of the product of two matrices is
the reverse product of their transposes. We thus have:
C
T
A = (A
T
C)
T = I
T = I. This shows that the matrix A also
has a left inverse C
T . The left and right inverses must be
equal, since C
T = C
T I = C
T AB = IB = B. Thus the matrix
B is indeed the full inverse matrix to A: AB = BA = I.
See also INVERSE MATRIX.
diagonal Any line joining two nonadjacent vertices
of a POLYGON is called a diagonal of the polygon. For
example, a square has two diagonals, each cutting the
figure into two congruent right-angled triangles, and a
pentagon has five different diagonals. There are no
diagonals in a triangle. In general, a regular ngon has
distinct diagonals.
A diagonal for a POLYHEDRON is any line joining
two vertices that are not in the same face. A cube, for
n(n – 3)
———–
2
1
det(A
–1 ) = ———
det(A)
128 diagonal