Class 12 Determinants · Long Answer3 Marksadjugate

Class 12 Maths · Determinants · Long Answer #94

For real square matrices A,B of the same order, prove adj(AB)=adj(B)adj(A), including singular cases.

✅ Step-by-Step Solution

Final Answer: adj(AB)=adj(B)adj(A)

If A and B are invertible, adj(M)=det(M)M⁻¹ gives adj(AB)=det(A)det(B)B⁻¹A⁻¹.
Regroup the scalar factors to obtain [det(B)B⁻¹][det(A)A⁻¹]=adj(B)adj(A). This proves the identity for invertible pairs.
For arbitrary A,B choose real t approaching 0 such that A+tI and B+tI are both invertible; only finitely many t are excluded by their determinant polynomials. Apply the proved identity to that pair. Every adjugate entry is a polynomial in the matrix entries, so taking t→0 preserves the equality.
Therefore the identity holds for all same-order real square matrices, including singular A or B.

[0.5 marks] If A and B are invertible, adj(M)=det(M)M⁻¹ gives adj(AB)=det(A)det(B)B⁻¹A⁻¹.
[1 mark] Regroup the scalar factors to obtain [det(B)B⁻¹][det(A)A⁻¹]=adj(B)adj(A). This proves the identity for invertible pairs.
[1 mark] For arbitrary A,B choose real t approaching 0 such that A+tI and B+tI are both invertible; only finitely many t are excluded by their determinant polynomials. Apply the proved identity to that pair. Every adjugate entry is a polynomial in the matrix entries, so taking t→0 preserves the equality.
[0.5 marks] Therefore the identity holds for all same-order real square matrices, including singular A or B.

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Determinants.