MATRICES AND VECTOR SPACES
may be shown that the rank of a generalM×Nmatrix is equal to the size of
the largest square submatrix ofAwhose determinant is non-zero. Therefore, if a
matrixAhas anr×rsubmatrixSwith|S|= 0, but no (r+1)×(r+ 1) submatrix
with non-zero determinant then the rank of the matrix isr. From either definition
it is clear that the rank ofAis less than or equal to the smaller ofMandN.
Determine the rank of the matrixA=
110 − 2
202 2
413 1
.
The largest possible square submatrices ofAmust be of dimension 3×3. Clearly,A
possesses four such submatrices, the determinants of which are given by
∣∣
∣
∣∣
∣
110
202
413
∣∣
∣
∣∣
∣
=0,
∣∣
∣
∣∣
∣
11 − 2
20 2
41 1
∣∣
∣
∣∣
∣
=0,
∣
∣∣
∣
∣∣
10 − 2
22 2
43 1
∣
∣∣
∣
∣∣=0,
∣
∣∣
∣
∣∣
10 − 2
02 2
13 1
∣
∣∣
∣
∣∣=0.
(In each case the determinant may be evaluated as described in subsection 8.9.1.)
The next largest square submatrices ofAare of dimension 2×2. Consider, for example,
the 2×2 submatrix formed by ignoring the third row and the third and fourth columns
ofA; this has determinant
∣
∣∣
∣
11
20
∣
∣∣
∣=1×^0 −^2 ×1=−^2.
Since its determinant is non-zero,Ais of rank 2 and we need not consider any other 2× 2
submatrix.
In the special case in which the matrixAis asquareN×Nmatrix, by comparingeither of the above definitions of rank with our discussion of determinants in
section 8.9, we see that|A|= 0 unless the rank ofAisN.Inotherwords,Ais
singularunlessR(A)=N.
8.12 Special types of square matrixMatrices that are square, i.e.N×N, are very common in physical applications.
We now consider some special forms of square matrix that are of particular
importance.
8.12.1 Diagonal matricesThe unit matrix, which we have already encountered, is an example of adiagonal
matrix. Such matrices are characterised by having non-zero elements only on the