Question

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Let A be the 2 × 2 matrix
([ a₁1 a₁2; a₂1 a₂2 ])
where a₁1, a₁2, a₂1, a₂2∈ℝ
find A⁻¹

Answer 1

To find the inverse of a 2 × 2 matrix A, calculate the determinant, and then use it to scale the adjugate of A. Ensure that the determinant is non-zero, otherwise, the inverse does not exist.

Step-by-step explanation:

To find the inverse of a 2 × 2 matrix A, given by A = ([a₁₁ a₁₂; a₂₁ a₂₂]) with a₁₁, a₁₂, a₂₁, a₂₂ ∈ ℝ, we use the formula A⁻¹ = (1/det(A)) × (adj(A)). The determinant of A, denoted as det(A), is calculated by a₁₁a₂₂ - a₁₂a₂₁. The adjugate of A, adj(A), is the transpose of the cofactor matrix of A, which for a 2 × 2 matrix is simply the matrix obtained by swapping the elements on the main diagonal and changing the signs of the off-diagonal elements. Therefore, the adjugate matrix of A is ([a₂₂ -a₁₂; -a₂₁ a₁₁]). The inverse of A is then: A⁻¹= (1/det(A)) × ([a₂₂ -a₁₂; -a₂₁ a₁₁]). Remember, a matrix has an inverse only if its determinant is non-zero. If det(A) = 0, A does not have an inverse.