inverse matrix

Question

The inverse of matrix A = [[2, 3], [1, 2]] is

RankGuru · Accepted Answer

1. **Determinant Calculation**: $|A| = (2)(2) - (3)(1) = 4 - 3 = 1$.
2. **Adjoint Method**: For a $2 	imes 2$ matrix $\begin{bmatrix} a & b \ c & d \end{bmatrix}$, the adjoint is $\begin{bmatrix} d & -b \ -c & a \end{bmatrix}$.
   - Swap main diagonal: $(2, 2) 	o (2, 2)$
   - Negate off-diagonal: $(3, 1) 	o (-3, -1)$
3. **Adjoint Result**: $adj(A) = \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}$.
4. **Apply Inverse Formula**: $A^{-1} = \frac{1}{|A|} adj(A) = \frac{1}{1} \begin{bmatrix} 2 & -3 \ -1 & 2 \end{bmatrix}$.

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