Jembi bought 2 kg of brand A and 3 kg of brand B for sh. 900. Amoit bought 1 kg of brand A and 2 kg of brand B for sh. 550. Find the cost of 1 kg of brand A and 1 kg of brand B.

Let $a$ be the cost of one kilogram of brand A rice.
Let $b$ be the cost of one kilogram of brand B rice.

From the problem statement, we can form two linear equations:
Jembi bought 2 kg of brand A and 3 kg of brand B for sh. 900:

$$2a + 3b = 900 \quad (1)$$

Amoit bought 1 kg of brand A and 2 kg of brand B for sh. 550:

$$a + 2b = 550 \quad (2)$$

Multiply equation (2) by 2 to make the coefficients of $a$ equal:

$$2 \times (a + 2b) = 2 \times 550$$ $$2a + 4b = 1100 \quad (3)$$

$$(2a + 4b) - (2a + 3b) = 1100 - 900$$ $$b = 200$$

$$a + 2(200) = 550$$ $$a + 400 = 550$$ $$a = 550 - 400$$ $$a = 150$$

The cost of one kilogram of brand A rice is sh. 150.
The cost of one kilogram of brand B rice is sh. 200.

The cost of one kilogram of brand A rice is $\boxed{\text{sh. 150}}$.
The cost of one kilogram of brand B rice is $\boxed{\text{sh. 200}}$.