Find the rank of matrix A = [[1, 2, 3, -4], [-2, 3, 7, -1], [1, 9, 16, -13]]

Steps to Solve

1. Set up the matrix and identify its entries

The given matrix ( A ) is formed by the entries:

$$ A = \begin{bmatrix} 1 & 2 & 3 & -4 \ -2 & 3 & 7 & -1 \ 1 & 9 & 16 & -13 \end{bmatrix} $$

2. Write the augmented matrix for row reduction

In order to find the rank, we can perform row operations on ( A ) until we reach its row echelon form.

3. Perform row operations to simplify the matrix

Start by using the first row to eliminate the first entry of the second row:

• Replace ( R_2 ) by ( R_2 + 2R_1 ):

$$ R_2 = R_2 + 2R_1 = [0, 7, 13, -9] $$

Next, simplify ( R_3 ):

• Replace ( R_3 ) by ( R_3 - R_1 ):

$$ R_3 = R_3 - R_1 = [0, 7, 13, -9] $$

Now, since ( R_2 ) and ( R_3 ) are the same, we can replace ( R_3 ) with ( R_2 ):

4. Continue row reduction steps

Continue row operations to reach echelon form:

• Replace ( R_3 ) with ( R_3 - R_2 ), resulting in:

$$ R_3 = [0, 0, 0, 0] $$

This shows that the third row is now all zeros.

5. Identify the rank

The non-zero rows (the first two rows) indicate that the rank of the matrix ( A ) is 2. Thus, the rank is determined by the count of non-zero rows.