To ensure that the matrix $A = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & a & 6 & 7 \\ 0 & 0 & 0 & b & 8 \end{bmatrix}$ has a rank of 2, we need to ensure that:

1. There are two linearly independent rows.
2. The third row must be linearly dependent on the first two rows.

For a matrix to have rank 2, the third row (in this case, $[0, 0, 0, b, 8]$) should be a linear combination of the first two rows.

Step 1: Analyze the third row conditions

The elements in the third row that determine linear independence are $b$ and $8$. For the third row to be dependent on the first two rows, we must set $b = 0$. If $b = 0$, then the third row is $[0, 0, 0, 0, 8]$, which is not a pivot row, hence making the matrix rank 2.

Step 2: Setting conditions for $a$

The element $a$ must not introduce a new pivot in the second row that would increase the rank of the matrix. Therefore, we can set $a = 0$.

Solution

To satisfy the condition that the matrix $A$ has a rank of 2, set: $a = 0 \quad \text{and} \quad b = 0$

The resulting matrix is: $A = \begin{bmatrix} 1 & 2 & 3 & 4 & 5 \\ 0 & 0 & 0 & 6 & 7 \\ 0 & 0 & 0 & 0 & 8 \end{bmatrix}$

With these values of $a$ and $b$, the matrix has a rank of 2.

Would you like more details on any step, or have any questions?

---

Here are some related questions to explore:

1. What would be the rank of the matrix if $a$ and $b$ were nonzero?
2. How do you determine if rows are linearly dependent in a matrix?
3. What is the general approach for finding the rank of a matrix?
4. Can the rank of a matrix change if we add a row or column?
5. How does the rank of a matrix affect solutions to a system of linear equations?

Tip: For matrices with parameters, you can often set specific entries to zero to control linear independence and determine rank.