Let's Remember — Chapter Summary

Let's Remember

• Order-3 determinant expansion (row 1): $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}=a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1(a_2b_3-a_3b_2)$.
• Minor/Cofactor: $M_{ij}$ = determinant left after deleting row $i$, column $j$; $C_{ij}=(-1)^{i+j}M_{ij}$.
• Seven properties of determinants: (i) transpose invariance, (ii) row/column swap flips sign, (iii) repeated row/column ⇒ value $0$, (iv) common factor of a row/column pulls outside, (v) a row/column that's a sum splits the determinant into a sum of two determinants, (vi) adding a multiple of one row/column to another leaves the value unchanged, (vii) triangle property — all-zero above/below the diagonal ⇒ value = product of diagonal entries.
• Cramer's Rule: $x=\dfrac{D_x}D,\ y=\dfrac{D_y}D,\ z=\dfrac{D_z}D$, valid when $D\ne0$.
• Consistency of three equations in $x,y$: $\begin{vmatrix}a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3\end{vmatrix}=0$ is necessary (not always sufficient).
• Area of a triangle: $A(\triangle)=\dfrac12\begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}$; the same determinant $=0$ tests collinearity.
• Scalar multiplication: $kA=[k\,a_{ij}]$.
• Addition: $A+B=[a_{ij}+b_{ij}]$, requires matching order. …