1. **Problem:** Find the value of $p$ such that the determinant of the matrix $$\begin{pmatrix}-2 & 1 & 1 \\ p & 1 & 3 \\ 1 & 2 & -1 \end{pmatrix} = 22$$ 2. **Formula:** The determinant of a $3 \times 3$ matrix $$\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ is given by $$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$$ 3. **Apply the formula:** Let $$a = -2, b = 1, c = 1$$ $$d = p, e = 1, f = 3$$ $$g = 1, h = 2, i = -1$$ Calculate each minor: $$ei - fh = (1)(-1) - (3)(2) = -1 - 6 = -7$$ $$di - fg = (p)(-1) - (3)(1) = -p - 3$$ $$dh - eg = (p)(2) - (1)(1) = 2p - 1$$ 4. **Substitute into determinant formula:** $$\det = -2(-7) - 1(-p - 3) + 1(2p - 1)$$ 5. **Simplify:** $$= 14 + p + 3 + 2p - 1$$ $$= (14 + 3 - 1) + (p + 2p)$$ $$= 16 + 3p$$ 6. **Set determinant equal to 22 and solve for $p$:** $$16 + 3p = 22$$ $$3p = 22 - 16$$ $$3p = 6$$ $$p = \frac{6}{3}$$ $$p = 2$$ **Final answer:** $p = 2$