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180 ALGEBRA

The determinant of the direct product of a matrix A of size m × m and B of size n × n
is calculated by the formula
m
n
det(A ⊗ B)= (det A) (det B) .
5.2.2-7. Relation between the determinant and the inverse matrix.

EXISTENCE THEOREM. A square matrix A is invertible if and only if its determinant is
different from zero.
Remark. A square matrix A is nondegenerate if its determinant is different from zero.
The adjugate (classical adjoint) for a matrix A ≡ [a ij ]of size n × n is a matrix C ≡ [c ij ]
T
of size n×n whose entries coincide with the cofactors of the entries of the transpose A , i.e.,
c ij = A ji (i, j = 1, 2, ... , n). (5.2.2.9)
The inverse matrix of a square matrix A ≡ [a ij ]of size n × n is the matrix of size n × n
obtained from the adjugate matrix by dividing all its entries by det A, i.e.,
A 11 A 21 A n1
⎛ ⎞
det A det A ··· det A
⎜ A 12 A 22 A n2 ⎟
–1 ⎜ det A det A ···
⎟ .
A = ⎜ . . det A ⎟ (5.2.2.10)
.
⎝ . . . . . . . . . ⎠
A 1n A 2n ··· A nn
det A det A det A
JACOBI THEOREM. For minors of the matrix of cofactors of a matrix A, the following
relations hold:
i 1
A i 1 A i 1 ··· A

j 1 j 2
j k
i 2 i 2
A A ··· A
i 2
j 1 j 2 j k =(det A) k–1 A i 1 i 2 ...i k .

. . . . j 1 j 2 ...j k
. . .
. . . .
.
A A ··· A
i k i k i k
j 1 j 2 j k
5.2.3. Equivalent Matrices. Eigenvalues
5.2.3-1. Equivalence transformation.
Matrices A and A of size m × n are said to be equivalent if there exist nondegenerate
2
matrices S and T of size m × m and n × n, respectively, such that A and A are related by
2
the equivalence transformation
A = SAT.
2
THEOREM. Two matrices of the same size are equivalent if and only if they are of the
same rank.
Remark 1. One of the square matrices S and T may coincide with the unit matrix. Thus, we have
equivalent matrices A and B if there is a nondegenerate square matrix S such that A = SA or A = AS.
2
2
Remark 2. Triangular decomposition of a matrix A corresponds to its equivalence transformation with
–1 –1 –1 –1
A ≡ I,sothat A = S T = LU,where L = S and P = T are an upper and lower triangular matrix. This
2
representation is also called the LU-decomposition.
Any equivalence transformation can be reduced to a sequence of elementary transfor-
mations of the following types:
1. Interchange of two rows (columns).
2. Multiplication of a row (column) by a nonzero scalar.
3. Addition to some row (column) of another row (column) multiplied by a scalar.