1. **State the problem:** We are given a 2x2 matrix $$B = \begin{bmatrix} 3 & 9 \\ 2 & x - 1 \end{bmatrix}$$ and we want to analyze or solve for properties involving this matrix. 2. **Common task:** A typical problem is to find the determinant of matrix $$B$$, which is important for understanding invertibility and other properties. 3. **Formula for determinant of a 2x2 matrix:** For $$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$, the determinant is $$\det(A) = ad - bc$$. 4. **Apply the formula to matrix $$B$$:** $$\det(B) = (3)(x - 1) - (9)(2)$$ 5. **Simplify the expression:** $$\det(B) = 3x - 3 - 18$$ $$\det(B) = 3x - 21$$ 6. **Interpretation:** The determinant depends on $$x$$. For example, if you want $$B$$ to be invertible, then $$\det(B) \neq 0$$, so: $$3x - 21 \neq 0$$ 7. **Solve for $$x$$ when determinant is zero:** $$3x - 21 = 0$$ $$3x = 21$$ $$x = 7$$ **Final answer:** The determinant of matrix $$B$$ is $$3x - 21$$. The matrix is invertible for all $$x \neq 7$$.