If adj B=A and P, Q are two unimodular matrices, i.e., |P|=1=|Q|, then (Q^-1 B P^-1)^-1 is equal

step 1 given adjoined of b equal to a and determinant of p equal to one equal to determinant of q we ne

If adj B=A and P, Q are two unimodular matrices, i.e.,...

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Key Concepts

Adjugate Matrix

The adjugate (or classical adjoint) of a matrix is closely related to its inverse. For an invertible matrix B, the adjugate is defined such that B?¹ = (1/det(B)) · adj(B). This relationship is particularly useful when the determinant of B is 1, as it then implies that B?¹ = adj(B). Recognizing this link is essential in transforming expressions involving inverses and adjugates within matrix equations.

Unimodular Matrices

Unimodular matrices are square matrices with a determinant equal to 1. This property is significant because it not only guarantees that the matrix is invertible but also that certain scale factors vanish from expressions involving determinants. In problems where unimodularity is assumed, the simplifying conditions aid in the analysis of transformations and inverses, as the determinant-based factors in formulas like the inverse of a matrix become one.

Inverse of a Product of Matrices

When dealing with the inverse of a product of matrices, a key property is that the inverse of a product is the product of the inverses in the reverse order. That is, for invertible matrices X, Y, and Z, (XYZ)?¹ = Z?¹Y?¹X?¹. This principle allows one to systematically dismantle composite expressions into simpler components, making it easier to manipulate and solve matrix equations.

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