1. **State the problem:** Find the determinant of the 4x4 matrix $$A=\begin{pmatrix}4 & -3 & 6 & -4 \\ 4 & -5 & -1 & -5 \\ -1 & -6 & -5 & 4 \\ -1 & -6 & 3 & -6\end{pmatrix}$$ using cofactor expansion. 2. **Formula and rules:** The determinant of a 4x4 matrix can be found by expanding along any row or column. We use the formula: $$\det(A) = \sum_{j=1}^4 (-1)^{i+j} a_{ij} M_{ij}$$ where $a_{ij}$ is the element in row $i$, column $j$, and $M_{ij}$ is the determinant of the $(4-1)\times(4-1)$ submatrix obtained by removing row $i$ and column $j$. 3. **Choose row for expansion:** We choose the first row for expansion: $$\det(A) = 4C_{11} + (-3)C_{12} + 6C_{13} + (-4)C_{14}$$ where $C_{ij} = (-1)^{i+j} M_{ij}$. 4. **Calculate minors:** - $M_{11}$ is determinant of $$\begin{pmatrix}-5 & -1 & -5 \\ -6 & -5 & 4 \\ -6 & 3 & -6\end{pmatrix}$$ - $M_{12}$ is determinant of $$\begin{pmatrix}4 & -1 & -5 \\ -1 & -5 & 4 \\ -1 & 3 & -6\end{pmatrix}$$ - $M_{13}$ is determinant of $$\begin{pmatrix}4 & -5 & -5 \\ -1 & -6 & 4 \\ -1 & -6 & -6\end{pmatrix}$$ - $M_{14}$ is determinant of $$\begin{pmatrix}4 & -5 & -1 \\ -1 & -6 & -5 \\ -1 & -6 & 3\end{pmatrix}$$ 5. **Calculate each minor determinant:** Use the formula for 3x3 determinant: $$\det = a(ei - fh) - b(di - fg) + c(dh - eg)$$ for matrix $$\begin{pmatrix}a & b & c \\ d & e & f \\ g & h & i\end{pmatrix}$$ - For $M_{11}$: $$a=-5, b=-1, c=-5, d=-6, e=-5, f=4, g=-6, h=3, i=-6$$ Calculate: $$ei - fh = (-5)(-6) - 4(3) = 30 - 12 = 18$$ $$di - fg = (-6)(-6) - 4(-6) = 36 + 24 = 60$$ $$dh - eg = (-6)(3) - (-5)(-6) = -18 - 30 = -48$$ So, $$\det(M_{11}) = -5(18) - (-1)(60) + (-5)(-48) = -90 + 60 + 240 = 210$$ - For $M_{12}$: $$a=4, b=-1, c=-5, d=-1, e=-5, f=4, g=-1, h=3, i=-6$$ Calculate: $$ei - fh = (-5)(-6) - 4(3) = 30 - 12 = 18$$ $$di - fg = (-1)(-6) - 4(-1) = 6 + 4 = 10$$ $$dh - eg = (-1)(3) - (-5)(-1) = -3 - 5 = -8$$ So, $$\det(M_{12}) = 4(18) - (-1)(10) + (-5)(-8) = 72 + 10 + 40 = 122$$ - For $M_{13}$: $$a=4, b=-5, c=-5, d=-1, e=-6, f=4, g=-1, h=-6, i=-6$$ Calculate: $$ei - fh = (-6)(-6) - 4(-6) = 36 + 24 = 60$$ $$di - fg = (-1)(-6) - 4(-1) = 6 + 4 = 10$$ $$dh - eg = (-1)(-6) - (-6)(-1) = 6 - 6 = 0$$ So, $$\det(M_{13}) = 4(60) - (-5)(10) + (-5)(0) = 240 + 50 + 0 = 290$$ - For $M_{14}$: $$a=4, b=-5, c=-1, d=-1, e=-6, f=-5, g=-1, h=-6, i=3$$ Calculate: $$ei - fh = (-6)(3) - (-5)(-6) = -18 - 30 = -48$$ $$di - fg = (-1)(3) - (-5)(-1) = -3 - 5 = -8$$ $$dh - eg = (-1)(-6) - (-6)(-1) = 6 - 6 = 0$$ So, $$\det(M_{14}) = 4(-48) - (-5)(-8) + (-1)(0) = -192 - 40 + 0 = -232$$ 6. **Calculate cofactors:** $$C_{11} = (-1)^{1+1} M_{11} = 1 \times 210 = 210$$ $$C_{12} = (-1)^{1+2} M_{12} = -1 \times 122 = -122$$ $$C_{13} = (-1)^{1+3} M_{13} = 1 \times 290 = 290$$ $$C_{14} = (-1)^{1+4} M_{14} = -1 \times (-232) = 232$$ 7. **Final determinant:** $$\det(A) = 4(210) + (-3)(-122) + 6(290) + (-4)(232)$$ Calculate each term: $$4 \times 210 = 840$$ $$-3 \times -122 = 366$$ $$6 \times 290 = 1740$$ $$-4 \times 232 = -928$$ Sum: $$840 + 366 + 1740 - 928 = 2018$$ **Answer:** $$\boxed{2018}$$