Consider the following in respect of two non-singular matrices A and B of the same order:

a. Det (A+ B) = det A + det B

b. (A + B)^-1 = A^-1 + B^-1

Which of the above is/are correct?

Concept:

A square matrix is nonsingular if its determinant is nonzero.

If  then determinant of A is given by:

|A| = (a­₁₁ × a₂₂) – (a₁₂ × a₂₁).

Note:

For 2 × 2 matrix,

Let A =  

than, Adj A =  

Note: Interchange the diagonal elements and change the sign of the remaining elements.

Calculation:

Let  and,

Now,

⇒ det A = (2 – 1) = 1

⇒ det B = (4 – 1) = 3

det A + det B = 1 + 3 = 4

⇒ det (A + B) = (12 – 4) = 8

∴ det (A+ B) ≠  det A + det B

Statement 1^st is wrong.

We know that for 2 × 2 matrix, interchange the diagonal elements and change the sign of remaining elements gives adjoint matrix.

We know that, 

Now,

∴ (A + B)^-1 ≠ A^-1 + B^-1

Statement 2^nd is wrong.