Find the rank of the matrix A = ⎡ 2 3 7 ⎤
⎣ -2 4 1 ⎦
⎢ -3 -1 ⎥ by reducing into echelon form.

To find the rank of the matrix A, we first need to reduce it to its echelon form. The given matrix A is:

A =
⎡ 2 3 7 ⎤
⎣ -2 4 1 ⎦
⎢ -3 -1 0 ⎥

Step 1: Form the Augmented Matrix

We can represent the matrix A as:

⎡ 2 3 7 ⎤
⎣ -2 4 1 ⎦
⎢ -3 -1 0 ⎥

Step 2: Row Operations

We will perform row operations to simplify the matrix:

• First, we can multiply the first row by 1/2 to make calculations easier:

R1 = (1/2) * R1 = ⎡ 1 1.5 3.5 ⎤

• Next, we can add 2 times the first row to the second row:

R2 = R2 + 2 * R1 = ⎡ 0 7 8 ⎤

• Then, we can add 3 times the first row to the third row:

R3 = R3 + 3 * R1 = ⎡ 0 3.5 10.5 ⎤

After these operations, the matrix looks like this:

⎡ 1 1.5 3.5 ⎤
⎣ 0 7 8 ⎦
⎢ 0 3.5 10.5 ⎥

Step 3: Further Simplification

Now, we can simplify the second and third rows:

• Divide the second row by 7:

R2 = (1/7) * R2 = ⎡ 0 1 1.14 ⎤

• Next, we can subtract 3.5 times the second row from the third row:

R3 = R3 - 3.5 * R2 = ⎡ 0 0 6.0 ⎤

The matrix now appears as:

⎡ 1 1.5 3.5 ⎤
⎣ 0 1 1.14 ⎦
⎢ 0 0 6.0 ⎥

Step 4: Determine the Rank

The echelon form of the matrix shows three non-zero rows. Therefore, the rank of matrix A is:

Rank(A) = 3