Determinants of 2x2 and 3x3 Matrices

Determinant of a 2x2 Matrix

Formula: For a 2x2 matrix \(\begin{pmatrix} A & B \ C & D \end{pmatrix}\), determinant is calculated as: \ AD - BC \
Example 1: \(\begin{pmatrix} 3 & 5 \ -4 & 7 \end{pmatrix}\)
- Calculation: \ 3 \times 7 - 5 \times (-4) = 21 + 20 = 41 \
Example 2: \(\begin{pmatrix} -7 & 8 \ 4 & -3 \end{pmatrix}\)
- Calculation: \ (-7) \times (-3) - 4 \times 8 = 21 - 32 = -11 \

Formula: For a 3x3 matrix \(\begin{pmatrix} A1 & A2 & A3 \ B1 & B2 & B3 \ C1 & C2 & C3 \end{pmatrix}\), determinant is: \ \ A1 \begin{vmatrix} B2 & B3 \ C2 & C3 \end{vmatrix} - B1 \begin{vmatrix} A2 & A3 \ C2 & C3 \end{vmatrix} + C1 \begin{vmatrix} A2 & A3 \ B2 & B3 \end{vmatrix} \
Example: \(\begin{pmatrix} 2 & 4 & -3 \ 5 & 7 & 6 \ -8 & 1 & 9 \end{pmatrix}\)
1. Step 1: Expand by first row
- \ 2 \begin{vmatrix} 7 & 6 \ 1 & 9 \end{vmatrix} - 4 \begin{vmatrix} 5 & 6 \ -8 & 9 \end{vmatrix} - 3 \begin{vmatrix} 5 & 7 \ -8 & 1 \end{vmatrix} \
1. Step 2: Calculate 2x2 determinants
- \ 2 \times (7\times9 - 6\times1) - 4 \times (5\times9 - (-8)\times6) - 3 \times (5\times1 - (-8)\times7) \
- \ 2 \times (63 - 6) - 4 \times (45 + 48) - 3 \times (5 + 56) \
- \ 2 \times 57 - 4 \times 93 - 3 \times 61 \
- Result: \ 2 \times 57 = 114 \
- \ -4 \times 93 = -372 \
- \ -3 \times 61 = -183 \ \ 114 - 372 - 183 = -441\
Example: \(\begin{pmatrix} 5 & 7 & -8 \ 4 & -3 & 6 \ 1 & 7 & -9 \end{pmatrix}\)
1. Step 1: Expand by first row
- \ 5 \begin{vmatrix} -3 & 6 \ 7 & -9 \end{vmatrix} - 7 \begin{vmatrix} 4 & 6 \ 1 & -9 \end{vmatrix} - 8 \begin{vmatrix} 4 & -3 \ 1 & 7 \end{vmatrix} \
1. Step 2: Calculate 2x2 determinants
- \ 5 \times (-27 - 42) - 7 \times (-36 - 6) - 8 \times (28 + 3) \
- \ 5 \times (-15) + 7 \times 42 - 8 \times 31 \
- Result: \ -75 + 294 - 248 = -29\

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