What is the rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns.

In linear algebra, the rank of a matrix is a fundamental concept that describes the dimension of the vector space spanned by its rows or columns. It is defined as the maximum number of linearly independent rows or columns in the matrix.

To find the rank of a matrix, we can use row operations to reduce the matrix to row echelon form or reduced row echelon form. The number of non-zero rows in the resulting matrix is the rank of the original matrix. For example, consider the matrix:

A =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{bmatrix}

We can use row operations to reduce A to row echelon form:

\begin{bmatrix}
1 & 2 & 3 \\
0 & -3 & -6 \\
0 & 0 & 0 \\
\end{bmatrix}

The number of non-zero rows in this matrix is 2, so the rank of A is 2.

Alternatively, we can also find the rank of a matrix by examining its determinant. The rank of a matrix is equal to the number of non-zero determinants of its submatrices. For example, consider the matrix:

B =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9 \\
\end{bmatrix}

The submatrices of B are:

\begin{bmatrix}
1 & 2 \\
4 & 5 \\
\end{bmatrix}

\begin{bmatrix}
1 & 3 \\
4 & 6 \\
\end{bmatrix}

\begin{bmatrix}
2 & 3 \\
5 & 6 \\
\end{bmatrix}

\begin{bmatrix}
4 & 5 \\
7 & 8 \\
\end{bmatrix}

\begin{bmatrix}
4 & 6 \\
7 & 9 \\
\end{bmatrix}

\begin{bmatrix}
5 & 6 \\
8 & 9 \\
\end{bmatrix}

The determinants of these submatrices are 1, -2, -3, -3, 6, and 0, respectively. There are 4 non-zero determinants, so the rank of B is 4