Consider the system of equations with the following augmented matrix, in which a and b are real numbers.(1 0 2 a^2 3| b)(0 a b 0 1| a)(0 0 0 a 0| b^2-1)(b) For values of a and b for which the system is consistent, find its general solution?

Answer 1

The given system of equations can be represented by the following augmented matrix:

⎡1 0 2 a² 3 | b ⎤

⎢0 a b 0 1 | a ⎥

⎢0 0 0 a 0 | b²-1⎥

For the system to be consistent, the third row of the augmented matrix must be all zeros. This means that a must be equal to 0 and b²-1 must also be equal to 0. Solving for b, we find that b = ±1.

Substituting these values of a and b into the first two rows of the augmented matrix, we get:

⎡1 0 2 0 3 | ±1⎤

⎢0 0 ±1 0 1 | 0 ⎥

From the second row, we can see that the third variable must be equal to ±1. Let’s call this variable x3. Substituting this value into the first row, we get:

1x1 + 2x3 + 3x5 = ±1

x1 + 2(±1) + 3x5 = ±1

x1 = -3x5 ∓ 2

Thus, the general solution to the system is given by:

x1 = -3x5 ∓ 2

x2 = free variable

x3 = ±1

x4 = free variable

x5 = free variable

where x2, x4, and x5 can take on any value.