ask ChatGPT for the correct rank algorithm

the prompt was
> how to compute the rank of a matrix in Rust? below is the structure of a matrix
>
> ```
> pub struct Matrix<T: Field> {
>     pub elements: Vec<T>,
>     pub height: usize,
>     pub width: usize,
> }
> ```

src/linalg.rs

@@ -165,34 +165,49 @@ impl<T: Field> Matrix<T> {
     }
 
     pub fn rank(&self) -> usize {
-        let mut matrix = self.clone();
+        let mut mat = self.elements.clone();
+        let mut rank = 0;
+        let mut i = 0;
 
-        for i in 0..matrix.height.min(matrix.width) {
-            let pivot = matrix.get(i, i);
-            if pivot.is_zero() {
-                continue;
+        for j in 0..self.width {
+            let mut found = false;
+            for k in i..self.height {
+                if mat[k * self.width + j] != T::zero() {
+                    // Swap the rows to move the non-zero element to the diagonal
+                    mat.swap(
+                        i * self.width..(i + 1) * self.width,
+                        k * self.width..(k + 1) * self.width,
+                    );
+                    found = true;
+                    break;
+                }
             }
 
-            matrix.divide_row_by(i, pivot);
+            if found {
+                for k in (i + 1)..self.height {
+                    let ratio = mat[k * self.width + j] / mat[i * self.width + j];
+                    for l in j..self.width {
+                        mat[k * self.width + l] -= ratio * mat[i * self.width + l];
+                    }
+                }
+                i += 1;
+            }
+        }
 
-            for k in 0..matrix.height {
-                if k != i {
-                    let factor = matrix.get(k, i);
-                    matrix.multiply_row_by_and_add_to_row(i, -factor, k);
+        for i in 0..self.height {
+            let mut all_zeros = true;
+            for j in 0..self.width {
+                if mat[i * self.width + j] != T::zero() {
+                    all_zeros = false;
+                    break;
                 }
             }
+            if !all_zeros {
+                rank += 1;
+            }
         }
 
-        matrix.height
-            - (0..matrix.height)
-                .map(|i| {
-                    let row =
-                        matrix.elements[(i * matrix.width)..((i + 1) * matrix.width)].to_vec();
-                    row.iter().all(|&x| x.is_zero())
-                })
-                .filter(|&r| r)
-                .collect::<Vec<_>>()
-                .len()
+        rank
     }
 
     pub(super) fn mul(&self, rhs: &Self) -> Result<Self, KomodoError> {