Determinants

Maths-2

Karthik Thiagarajan

IIT Madras

Geometry (2D)

\[ \begin{vmatrix} 1 & 2\\ 3 & 4 \end{vmatrix} = 1 \times 4 - 2 \times 3 = -2 \]

\[ \begin{vmatrix} a & b\\ c & d \end{vmatrix} = ad - bc \]

\[ \begin{vmatrix} 1 & 0\\ 0 & 1 \end{vmatrix} = 1 \]

\[ \begin{vmatrix} 2 & 0\\ 0 & 3 \end{vmatrix} = 6 \]

\[ \begin{vmatrix} l & 0\\ 0 & b \end{vmatrix} = lb \]

Geometry (3D)

\[ \small \begin{aligned} \begin{vmatrix} 1 & 2 & 3\\ -1 & 0 & 2\\ 3 & 1 & 2 \end{vmatrix} &= 1 \times (0 \times 2 - 1 \times 2) +\\ &= (-2) \times (-1 \times 2 - 3 \times 2) +\\\\ &= 3 \times (-1 \times 1 - 3 \times 0)\\\\ &= 11 \end{aligned} \]

\[ \begin{aligned} \begin{vmatrix} a & b & c\\ d & e & f\\ g & h & i \end{vmatrix} &= a (ei - hf) +\\ &= -b(di - gf) +\\\\ &= + c(dh - ge) \end{aligned} \]

\[ \begin{vmatrix} a & 0 & 0\\ 0 & b & 0\\ 0 & 0 & c \end{vmatrix} = abc \]

Geometry (general)

\[ A = \begin{bmatrix} a & b\\ c & d \end{bmatrix} \]

\[ \text{Area} = |\text{det}(A)| \]

\[ A = \begin{bmatrix} a & b & c\\ d & e & f\\ g & h & i \end{bmatrix} \]

\[ \text{Volume} = |\text{det}(A)| \]