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Unit-4(Theorems)

Determinants

**Theorem-1 :-** If A be any given square matrix of order n, then A(adj A) = (adj A) A = |A|I , where I is the identity matrix of order n

Proof :

**Theorem-2 :-** A square matrix A is invertible if and only if A is nonsingular matrix.

Proof : Let A be invertible matrix of order n and I be the identity matrix of order n. Then, there exists a square matrix B of order n such that AB = BA = I Now AB = I. So |AB| =|I| or |A||B| = 1 (since |I|=1, |AB|= |A||B|) This gives |A| ≠ 0. Hence A is nonsingular. Conversely, let A be nonsingular. Then |A| ≠ 0 Now A (adj A) = (adj A) A = |A| I (Theorem 1) A[adj(A)/|A|] = [adj(A)/|A|]A = I or AB = BA = I, where B = adj(A)/|A| Thus A is invertible and A^{-1}= 1/|A| X adj(A).

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