04Class 12 Maths

Determinants formulas

Determinant, adjoint and inverse formulas, including solving linear equations. Expect at least one calculation you can finish faster with these properties.

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All Maths formulas

Determinants

2 × 2 determinant: $\displaystyle \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$
Cofactor: $\displaystyle A_{ij} = (-1)^{i + j}M_{ij}$
Scalar multiple: $\displaystyle |kA| = k^n|A|$
n is the order of A.
Product and transpose: $\displaystyle |AB| = |A||B|, \quad |A'| = |A|$
Area of a triangle: $\displaystyle \Delta = \frac12\begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}$
Take the absolute value.

Adjoint and inverse

Adjoint: $\displaystyle A(\text{adj}A) = (\text{adj}A)A = |A|I$
Inverse: $\displaystyle A^{-1} = \frac{1}{|A|}\text{adj}A$
Exists only when |A| ≠ 0.
Adjoint's determinant: $\displaystyle |\text{adj}A| = |A|^{n - 1}$

Solving equations

Matrix method: $\displaystyle AX = B \;\Rightarrow\; X = A^{-1}B$

Where students slip up

Writing |kA| = k|A|. For an order-n matrix, it's kⁿ|A|.
Forgetting the sign pattern when finding cofactors.
Concluding 'no solution' straight away when |A| = 0. Check (adj A)B to tell no solution from infinitely many.

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