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

Steps to Solve

1. Write down the matrix A

The given matrix A is:

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

2. Row Reduction

We need to perform row operations to reduce the matrix to Row Echelon Form (REF).

• First, we will swap Row 1 and Row 2 for convenience:

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

• Next, we will scale Row 1 by multiplying it by $-\frac{1}{2}$:

$$ R_1 \leftarrow -\frac{1}{2} R_1 $$

Resulting in:

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

3. Eliminate the first column below Row 1

Now we eliminate the elements below the leading 1 in Column 1:

• For Row 2: $$ R_2 \leftarrow R_2 - 2R_1 $$

• For Row 3: $$ R_3 \leftarrow R_3 - 1R_1 $$

Resulting in:

$$ A = \begin{bmatrix} 1 & -\frac{3}{2} & -\frac{7}{2} \ 0 & 6 & 3 \ 0 & \frac{21}{2} & -\frac{13}{2} \end{bmatrix} $$

4. Continue Row Reduction

Now we simplify Row 2:

• Scale Row 2 by multiplying it by $\frac{1}{6}$:

$$ R_2 \leftarrow \frac{1}{6} R_2 $$

Resulting in:

$$ A = \begin{bmatrix} 1 & -\frac{3}{2} & -\frac{7}{2} \ 0 & 1 & \frac{1}{2} \ 0 & \frac{21}{2} & -\frac{13}{2} \end{bmatrix} $$

5. Eliminate the second column below Row 2

Eliminate the entry below Row 2:

• For Row 3: $$ R_3 \leftarrow R_3 - \frac{21}{2} R_2 $$

Resulting in:

$$ A = \begin{bmatrix} 1 & -\frac{3}{2} & -\frac{7}{2} \ 0 & 1 & \frac{1}{2} \ 0 & 0 & -\frac{17}{2} \end{bmatrix} $$

6. Final Row Echelon Form and Count Rows

Now our matrix is in Row Echelon Form, and we see that there are three non-zero rows. Thus, the rank of the matrix is 3.