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5.2. MATRICES AND DETERMINANTS 177

3. Linearity with respect to a row (or column) of the corresponding matrix: suppose
that the ith row of a matrix A ≡ [a ij ] is a linear combination of two row vectors,
(a i1 , ... , a i3 )= λ(b 1 , ... , b n )+ μ(c 1 , ... , c n ); then
det A = λ det A b + μ det A c ,
where A b and A c are the matrices obtained from A by replacing its ith row with
(b 1 , ... , b n )and (c 1 , ... , c n ). This fact, together with the first property, implies that a
similar linearity relation holds if a column of the matrix A is a linear combination of
two column vectors.

Some useful corollaries from the basic properties:
1. The determinant of a matrix with two equal rows (columns) is equal to zero.
2. If all entries of a row are multiplied by λ, the determinant of the resulting matrix is
multiplied by λ.
3. If a matrix contains a row (columns) consisting of zeroes, then its determinant is equal
to zero.
4. If a matrix has two proportional rows (columns), its determinant is equal to zero.
5. If a matrix has a row (column) that is a linear combination of its other rows (columns),
its determinant is equal to zero.
6. The determinant of a matrix does not change if a linear combination of some of its rows
is added to another row of that matrix.
THEOREM (NECESSARY AND SUFFICIENT CONDITION FOR A MATRIX TO BE DEGENER-
ATE). The determinant of a square matrix is equal to zero if and only if its rows (columns)
are linearly dependent.

5.2.2-3. Minors. Basic minors. Rank and defect of a matrix.
Let A ≡ [a ij ]beamatrix of size n × n.Its mth-order (m ≤ n) minor of the first kind,
denoted by M i 1 i 2 ...i m , is the mth-order determinant of a submatrix obtained from A by
j 1 j 2 ...j m
removing some of its n – m rows and n – m columns. Here, i 1 , i 2 , ... , i m are the
indices of the rows and j 1 , j 2 , ... , j m are the indices of the columns involved in that
i 1 i 2 ...i m
submatrix. The (n – m)th-order determinant of the second kind, denoted by M ,is
j 1 j 2 ...j m
the (n – m)th-order determinant of the submatrix obtained from A by removing the rows
and the columns involved in M i 1 i 2 ...i m .The cofactor of the minor M i 1 i 2 ...i m is defined by
j 1 j 2 ...j m j 1 j 2 ...j m
A i 1 i 2 ...i m =(–1) i 1 +i 2 +···+i m+j 1 +j 2 +···+j m M i 1 i 2 ...i m .
j 1 j 2 ...j m j 1 j 2 ...j m
Remark. minors of the first kind can be introduced for any rectangular matrix A ≡ [a ij]of size m × n.Its
kth-order (k ≤ min{m, n}) minor M i 1 i 2 ...i k is the determinant of the submatrix obtained from A by removing
j 1 j 2 ...j k
some of its m – k rows and n – k columns.
LAPLACE THEOREM. Given m rows with indices i 1 , ... , i m (or m columns with indices
j 1 , ... , j m )ofa square matrix A, its determinant Δ is equal to the sum of products of all
mth-order minors M i 1 i 2 ...i m in those rows (resp., columns) and their cofactors A i 1 i 2 ...i m , i.e.,
j 1 j 2 ...j m j 1 j 2 ...j m

Δ ≡ det A = M i 1 i 2 ...i m A i 1 i 2 ...i m = M i 1 i 2 ...i m A i 1 i 2 ...i m .
j 1 j 2 ...j m j 1 j 2 ...j m j 1 j 2 ...j m j 1 j 2 ...j m
j 1 ,j 2 ,...,j m i 1 ,i 2 ,...,i m
Here, in the first sum i 1 , ... , i m are fixed, and in the second sum j 1 , ... , j m are fixed.
Let A ≡ [a ij ] beamatrix ofsize m × n with at least one nonzero entry. Then there is a
positive integer r ≤ n for which the following conditions hold:
i) the matrix A has an rth-order nonzero minor, and
ii) any minor of A of order (r + 1) and higher (of it exists) is equal to zero.