1. **Problem 1: Find the determinant of matrix A** Given matrix $$A=\begin{bmatrix}2 & 3 & 5 \\ 4 & 1 & 6 \\ 1 & 4 & 0\end{bmatrix}$$ 2. **Formula: Determinant of a 3x3 matrix** $$\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where the matrix is $$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$ 3. **Apply the formula to matrix A:** $$a=2, b=3, c=5, d=4, e=1, f=6, g=1, h=4, i=0$$ Calculate each term: - $$ei - fh = (1)(0) - (6)(4) = 0 - 24 = -24$$ - $$di - fg = (4)(0) - (6)(1) = 0 - 6 = -6$$ - $$dh - eg = (4)(4) - (1)(1) = 16 - 1 = 15$$ 4. **Substitute back:** $$\det(A) = 2(-24) - 3(-6) + 5(15) = -48 + 18 + 75 = 45$$ --- 5. **Problem 2: Using Sarrus rule, evaluate determinant of matrix B** Given matrix $$B=\begin{bmatrix}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{bmatrix}$$ 6. **Sarrus rule for 3x3 matrix:** Sum of products of diagonals from left to right minus sum of products of diagonals from right to left. Calculate: - Left to right diagonals: $$1 \times 5 \times 9 = 45$$ $$2 \times 6 \times 7 = 84$$ $$3 \times 4 \times 8 = 96$$ Sum = $$45 + 84 + 96 = 225$$ - Right to left diagonals: $$3 \times 5 \times 7 = 105$$ $$1 \times 6 \times 8 = 48$$ $$2 \times 4 \times 9 = 72$$ Sum = $$105 + 48 + 72 = 225$$ 7. **Determinant:** $$\det(B) = 225 - 225 = 0$$ --- 8. **Problem 3: Find k in the equation** $$1x^2 + 4x + 11 = (2 + 3x)^2 - 4(2x - 1) = k$$ 9. **Expand and simplify:** $$(2 + 3x)^2 = 4 + 12x + 9x^2$$ $$-4(2x - 1) = -8x + 4$$ Sum: $$4 + 12x + 9x^2 - 8x + 4 = 9x^2 + (12x - 8x) + (4 + 4) = 9x^2 + 4x + 8$$ 10. **Equate to left side:** $$1x^2 + 4x + 11 = k = 9x^2 + 4x + 8$$ Since the expressions are equal for all x, coefficients must match: - For $$x^2$$: $$1 = 9$$ (not equal) - For $$x$$: $$4 = 4$$ (equal) - Constants: $$11 = 8$$ (not equal) Therefore, no single $$k$$ satisfies this equality for all $$x$$. --- **Final answers:** - $$\det(A) = 45$$ - $$\det(B) = 0$$ - The expression for $$k$$ is $$9x^2 + 4x + 8$$, which does not equal $$1x^2 + 4x + 11$$ for all $$x$$.