Determinants — chapter notes

Every square matrix $A$ has a number called its determinant, written $|A|$ or $\det A$. A non-square matrix has no determinant.

• Order 1: $|[a]| = a$.
• Order 2: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$.

Example: $\begin{vmatrix} 5 & -3 \\ 4 & 2 \end{vmatrix} = 10 - (-12) = 22$. Keep negative products in brackets to avoid sign slips.

Choose any row (or column). For each element $a_{ij}$ in it, delete row $i$ and column $j$; the $2 \times 2$ determinant left is the minor $M_{ij}$. Attach the sign $(-1)^{i + j}$ (a chessboard of $+$ and $-$ starting with $+$ at the top left), multiply by the element and add.

Along row 1: $|A| = a_{11}M_{11} - a_{12}M_{12} + a_{13}M_{13}$.

Any row or column gives the same value. Choose the one with the most zeros. Example: $\begin{vmatrix} 2 & -1 & 3 \\ 1 & 4 & 0 \\ -2 & 1 & 5 \end{vmatrix} = 72$ along row 1 ($40 + 5 + 27$) and along column 3 ($27 + 0 + 45$).

Equations: evaluate both sides, simplify, and give all roots — e.g. $\begin{vmatrix} x & 4 \\ x & 2x \end{vmatrix} = 16$ gives $x = 4$ or $x = -2$.

Not in the 2026–27 syllabus: properties of determinants (row and column operations). Evaluate by direct expansion.

For vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, the area is $\frac12|\Delta|$ where $\Delta = \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$. Always take the absolute value: $\Delta$ may be negative.

Given the area: $\Delta = \pm 2 \times \text{area}$ — two equations, often two answers (e.g. $k = 0$ or $8$).

Collinearity: three points are collinear exactly when $\Delta = 0$ (zero area). Example: $(1, -1)$, $(3, 3)$, $(k, 7)$ are collinear for $k = 5$.

Line through two points: a general point $(x, y)$ is on the line through $(x_1, y_1)$ and $(x_2, y_2)$ exactly when $\begin{vmatrix} x & y & 1 \\ x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \end{vmatrix} = 0$. Through $(2, 1)$ and $(4, 5)$: $y = 2x - 3$.

• Minor $M_{ij}$: delete row $i$, column $j$, evaluate.
• Cofactor $A_{ij} = (-1)^{i + j}M_{ij}$.
• Expansion with cofactors: $|A| = a_{i1}A_{i1} + a_{i2}A_{i2} + a_{i3}A_{i3}$ for any row $i$.
• Adjoint: $\operatorname{adj} A$ is the transpose of the matrix of cofactors. Its first row holds the cofactors of the first column of $A$.
• $2 \times 2$ shortcut: $\operatorname{adj}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.

Example: for $Q = \begin{bmatrix} 2 & 1 & 3 \\ 1 & -1 & 0 \\ 0 & 2 & 1 \end{bmatrix}$, $\operatorname{adj} Q = \begin{bmatrix} -1 & 5 & 3 \\ -1 & 2 & 3 \\ 2 & -4 & -3 \end{bmatrix}$.

Key identity: $A(\operatorname{adj} A) = (\operatorname{adj} A)A = |A|I$.

• $A$ is singular if $|A| = 0$ and non-singular if $|A| \ne 0$.
• $A$ is invertible if and only if it is non-singular, and then $A^{-1} = \dfrac{1}{|A|}\operatorname{adj} A$.

Method: $|A|$ → cofactors → adjoint (transpose!) → divide by $|A|$ → check $AA^{-1} = I$.

Example: $A = \begin{bmatrix} 6 & 2 \\ 5 & 3 \end{bmatrix}$, $|A| = 8$, $A^{-1} = \frac18\begin{bmatrix} 3 & -2 \\ -5 & 6 \end{bmatrix}$.

Not in the 2026–27 syllabus: finding the inverse by elementary row operations.

For a square matrix $A$ of order $n$:

• $|\operatorname{adj} A| = |A|^{n - 1}$ (order 3: $|A|^2$).
• $|kA| = k^n|A|$ (order 3: $k^3|A|$).
• $|A^{-1}| = \dfrac{1}{|A|}$, from $|A||A^{-1}| = |I| = 1$.
• $|AB| = |A||B|$ for square matrices of the same order.

Example: order $3$, $|A| = 3$: $|2\operatorname{adj} A| = 2^3 \times 3^2 = 72$.

Write the system as $AX = B$ (coefficients in $A$ row by row, unknowns in $X$, constants in $B$).

• $|A| \ne 0$: consistent, unique solution $X = A^{-1}B = \dfrac{1}{|A|}(\operatorname{adj} A)B$.
• $|A| = 0$, $(\operatorname{adj} A)B \ne O$: inconsistent — no solution.
• $|A| = 0$, $(\operatorname{adj} A)B = O$: infinitely many solutions or none; check the equations directly.

Example: $3x + 2y = 7$, $5x - 4y = 19$ gives $x = 3$, $y = -1$. Example: $x + y + z = 4$, $2x - y + z = 8$, $x + 2y - z = -3$ gives $(2, -1, 3)$. Always substitute back.