1. **State the problem:** We want to find the coefficient of $x^{-5}$ in the expansion of $\left(2x - 3x^{-2}\right)^7$ and determine the value of $k$ corresponding to that term in the binomial expansion. 2. **Recall the binomial expansion formula:** $$\left(a + b\right)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ 3. **Identify terms:** Here, $a = 2x$, $b = -3x^{-2}$, and $n=7$. 4. **General term:** The $k$-th term (starting from $k=0$) is $$T_{k+1} = \binom{7}{k} (2x)^{7-k} (-3x^{-2})^k = \binom{7}{k} 2^{7-k} (-3)^k x^{7-k} x^{-2k} = \binom{7}{k} 2^{7-k} (-3)^k x^{7 - k - 2k} = \binom{7}{k} 2^{7-k} (-3)^k x^{7 - 3k}$$ 5. **Find $k$ such that the power of $x$ is $-5$:** $$7 - 3k = -5$$ $$7 + 5 = 3k$$ $$12 = 3k$$ $$k = 4$$ 6. **Calculate the coefficient for $k=4$:** $$\binom{7}{4} 2^{7-4} (-3)^4 = \binom{7}{4} 2^3 3^4$$ 7. **Calculate each part:** $$\binom{7}{4} = \frac{7!}{4!3!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35$$ $$2^3 = 8$$ $$3^4 = 81$$ 8. **Combine:** $$35 \times 8 \times 81 = 35 \times 648 = 22680$$ 9. **Sign:** Since $(-3)^4$ is positive, the coefficient is positive. **Final answer:** The coefficient of $x^{-5}$ is $22680$ and it corresponds to $k=4$ in the expansion.