KCET2026MathematicsDeterminants
If A and B are invertible square matrices of order n, then which of the following is not correct?
Options
- A(AB) = (A) (B)
- B(kA) = k^n (A)
- C(A + B) = (A) + (B)
- D(A⁻¹) = 1 (A)
Correct answer
C. (A + B) = (A) + (B)
Step-by-step solution
For any two square matrices A and B of the same order, the determinant of their product is the product of their determinants, so (AB) = (A) (B) . If A is a square matrix of order n and k is a scalar, then multiplying A by k multiplies each of its n rows by k . Thus, (kA) = k^n (A) . For an invertible matrix A , A A⁻¹ = I . Taking the determinant on both sides gives (A) (A⁻¹) = (I) = 1 , which implies (A⁻¹) = 1 (A) . However, the determinant of a sum of matrices is generally not equal to the sum of their determinant