Chapter 4 at a Glance

The determinant

Every square matrix $A$ carries one number $\det A = \vert A \vert$. For order $1$: $\vert [a] \vert = a$. For order $2$: $\begin{vmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{vmatrix} = a_{11}a_{22} - a_{12}a_{21}$. For order $3$: expand along any row or column — signed elements times $2 \times 2$ minors — all six choices agree; pick the one with the most zeros.

Minor $M_{ij}$: delete row $i$, column $j$. Cofactor $A_{ij} = (-1)^{i+j}M_{ij}$. Then:

1. elements $\times$ own cofactors $= \Delta$ (any row or column),
2. elements $\times$ another row's cofactors $= 0$.

Area and collinearity

$$\Delta = \frac{1}{2}\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$$

Take the absolute value; when the area is given, equate to $\pm$ the area; collinear points $\Leftrightarrow$ determinant $= 0$ (which also produces the equation of a line through two points).

Adjoint and inverse

$\operatorname{adj} A$ = transpose of the cofactor matrix ($2 \times 2$ shortcut: swap the diagonal, negate the off-diagonal). The master identity and its offspring:

$$A(\operatorname{adj} A) = (\operatorname{adj} A)A = \vert A \vert I, \qquad A^{-1} = \frac{1}{\vert A \vert}\operatorname{adj} A \quad (\vert A \vert \neq 0)$$

Singular: $\vert A \vert = 0$ (no inverse). Nonsingular: $\vert A \vert \neq 0$ (invertible). Key formulas:

Solving systems

Write the system as $AX = B$. If $\vert A \vert \neq 0$: unique solution $X = A^{-1}B$. If $\vert A \vert = 0$: compute $(\operatorname{adj} A)B$ — nonzero means inconsistent; zero means infinitely many solutions or none.

Mistake checklist

1. $\vert A \vert$ is not a modulus — determinants can be negative, and only square matrices have them.
2. $\vert kA \vert = k\vert A \vert$ is wrong — the correct factor is $k^n$, one per row.
3. Cofactor $\neq$ minor — the $(-1)^{i+j}$ flip applies whenever $i + j$ is odd.
4. Adjoint needs the transpose — cofactor matrix first, then flip across the diagonal.
5. Given-area problems need $\pm$ — solving with one sign loses half the answers.
6. $X = A^{-1}B$, never $BA^{-1}$ — premultiply, and state $\vert A \vert \neq 0$ before inverting.
7. $\vert A + B \vert \neq \vert A \vert + \vert B \vert$ — determinants respect products, not sums.
8. Skew-symmetric zero rule needs odd order — at even order the determinant can be nonzero.

The 15 questions below are a fast pass over the whole chapter at recall level. Each explanation names the section to revisit if it feels shaky.