Question

For which values of a, b in R is the system with matrix [a, 2; b, 3] consistent?

Answer 1

The system with matrix [a, 2; b, 3] is consistent for all values of a and b in real numbers (R).

Step-by-step explanation:

The system with matrix [a, 2; b, 3] is consistent for all values of a and b in real numbers (R).

To prove this, we need to show that the system does not have any inconsistent equations or contradictions. To do this, we can set up the augmented matrix and perform row operations to determine if the system has a unique solution or infinitely many solutions.

Let's start:

1. Augmented Matrix: [a, 2 | b]
2. Row 2 = Row 2 - (b/a) * Row 1
3. A new augmented matrix is formed: [a, 2 | b - (2*b/a)]
4. If a is not equal to 0, the system will have a unique solution.
5. If a = 0 and b is not equal to 0, the system will be inconsistent and have no solution.
6. If both a and b are equal to 0, the system will have infinitely many solutions.

Therefore, the system is consistent for all values of a and b in R.