KCET · Maths · Matrices
If \(A\) and \(B\) are invertible matrices then which of the following is not correct ?
- A \(\operatorname{adj} A=|A| A^{-1}\)
- B \(\operatorname{det}\left(A^{-1}\right)=[\operatorname{det}(A)]^{-1}\)
- C \((A B)^{-1}=B^{-1} A^{-1}\)
- D \((A+B)^{-1}=B^{-1}+A^{-1}\)
Answer & Solution
Correct Answer
(D) \((A+B)^{-1}=B^{-1}+A^{-1}\)
Step-by-step Solution
Detailed explanation
If \(A\) and \(B\) are invertible matrices, then
(a) \((A B)^{-1}=B^{-1} A^{-1}\)
(b) \(\operatorname{adj} A=|A| \cdot A^{-1}\)
(c) \(\left|A^{-1}\right|=|A|^{-1}\)
Hence, \((A+B)^{-1} \neq B^{-1}+A^{-1}\)
(a) \((A B)^{-1}=B^{-1} A^{-1}\)
(b) \(\operatorname{adj} A=|A| \cdot A^{-1}\)
(c) \(\left|A^{-1}\right|=|A|^{-1}\)
Hence, \((A+B)^{-1} \neq B^{-1}+A^{-1}\)
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