1.13. Using the method of bordering minors, find the rank of the matrix

$$
A=\left(\begin{array}{ccccc}
2 & 1 & 2 & 1 & 2 \\
1 & 1 & 5 & -2 & 3 \\
-1 & 0 & -4 & 4 & 1 \\
3 & 3 & 8 & 1 & 9
\end{array}\right)
$$
Solution. The second-order minor

$$
M_{2}=\left|\begin{array}{ll}
2 & 1 \\
1 & 1
\end{array}\right|=2 \cdot 1-1 \cdot 1=1 \neq 0
$$

is non-zero and located in the upper left corner of matrix $A$. The third-order minor

$$
M_{3}=\left|\begin{array}{ccc}
2 & 1 & 2 \\
1 & 1 & 5 \\
-1 & 0 & 4
\end{array}\right|=\left|\begin{array}{ccc}
1 & 0 & -3 \\
1 & 1 & 5 \\
-1 & 0 & 4
\end{array}\right|=\left|\begin{array}{cc}
1 & -3 \\
-1 & 4
\end{array}\right|=4-3=1 \neq 0
$$

enclosing the minor $M_{2}$, is also non-zero. However, both fourth-order minors enclosing $M_{3}$,

$$
M_{4}=\left|\begin{array}{cccc}
2 & 1 & 2 & 1 \\
1 & 1 & 5 & -2 \\
-1 & 0 & -4 & 4 \\
3 & 3 & 8 & 1
\end{array}\right|, M_{4}=\left|\begin{array}{cccc}
2 & 1 & 2 & 2 \\
1 & 1 & 5 & 3 \\
-1 & 0 & -4 & 1 \\
3 & 3 & 8 & 9
\end{array}\right|
$$

are equal to zero. Therefore, the rank of matrix $A$ is 3, and a basis minor is, for example, the minor $M$ presented above.

The method of elementary transformations is based on the fact that elementary transformations of a matrix do not change its rank. Using these transformations, the matrix can be brought to a form where all its elements, except for $a_{11}, a_{22}, \ldots$, $a_{r r}(r \leqslant \min (m, n))$, are zero. This obviously means that rang $A=r$. Note that if an $n$-order matrix has the form of an upper triangular matrix, i.e., a matrix where all elements below the main diagonal are zero, then its determinant is equal to the product of the elements on the main diagonal. This property can be used when calculating the rank of a matrix using the method of elementary transformations: it is necessary to use them to bring the matrix to a triangular form, and then, by isolating the corresponding determinant, we find that the rank of the matrix is equal to the number of elements on the main diagonal that are non-zero.