Rank of a matrix by means of determinants

The rank of a matrix can also be calculated using determinants. We can define rank using what interests us now.

The rank of a matrix is the order of the largest non-zero square submatrix.

See the following example.

A=\left( \begin{array}{ccccc} 2 & 1 & 3 & 2 & 0 \\ 3 & 2 & 5 & 1 & 0 \\ -1 & 1 & 0 & -7 & 0 \\ 3 & -2 & 1 & 17 & 0 \\ 0 & 1 & 1 & -4 & 0 \end{array} \right)

1) Given A , we eliminate rows or columns acording to the criterion to calculate the rank using the Gaussian elimination method. Thus,

Column 5 can be discarded because all its elements are zero.

Column 3 can be discarded because it is a linear combination of column 1 and column 2 . Specifically, c3=c1+c2 .

A=\left( \begin{array}{ccc} 2 & 1 & 2 \\ 3 & 2 & 1 \\ -1 & 1 & -7 \\ 3 & -2 & 17 \\ 0 & 1 & -4 \end{array} \right)

2) Is there any non-zero square submatrix of order 1 ?

Any non-zero element is a non-zero square submatrix, therefore we will look at those of higher order.

Is there any non-zero square submatrix of order 2 ?

\left| \begin{array}{cc} 2 & 1 \\ 3 & 2 \end{array} \right| = 1 \neq 0

Yes, there is, therefore we will look for higher orders.

4) Is there any non-zero square submatrix of order 3 ?

\left| \begin{array}{ccc} 2 & 1 & 2 \\ 3 & 2 & 1 \\ -1 & 1 & -7 \end{array} \right| = 0

\left| \begin{array}{ccc} 3 & 2 & 1 \\ -1 & 1 & -7 \\ 3 & -2 & 17 \end{array} \right| = 0

\left| \begin{array}{ccc} -1 & 1 & -7 \\ 3 & -2 & 17 \\ 0 & 1 & -4 \end{array} \right| = 0

No, there is not. Therefore, rank(A)=2 , which is the order of the largest non-zero square submatrix.