Academia Aeternum
तमसो मा ज्योतिर्गमय
Wisdom · Knowledge · Eternity
Properties, Cofactors & Cramer's Rule — Every Determinant Exercise Solved
6 exercise files · 68 total questions
\(|A| = a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13}\)\(\text{Area} = \tfrac12\begin{vmatrix}x_1&y_1&1\x_2&y_2&1\x_3&y_3&1\end{vmatrix}\)\(A^{-1} = \dfrac{1}{|A|}\,\mathrm{adj}(A),\quad |A|\neq 0\)\(AX=B \Rightarrow X = A^{-1}B\ (\text{if } |A|\neq0)\)Step 1 — Simplify determinants first using row/column operations (Rᵢ→Rᵢ+kRⱼ) before expanding — never expand a messy determinant directly. Step 2 — For area: take absolute value; if the determinant is 0, points are collinear. Step 3 — Adjoint: compute the full cofactor matrix carefully with sign pattern, then transpose. Step 4 — Matrix method for equations: verify |A|≠0 before computing A⁻¹; state 'no solution' or 'infinite solutions' if |A|=0.
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