Question

If A and B are invertible matrices, then which one of the following is not correct?
(a) adj A = $\left|A\right|$ A^-1 (b) det (A^-1) = [det(A)]^-1
(c) (AB)^-1 = B^-1A^-1 (d) (A+B)^-1 = B^-1 + A^-1

Answer 1

(a) adj A = $\left|A\right|$ A⁻¹

As we know,
$$\begin{gathered} A^{-1}=\frac{\mathrm{adj}A}{\left|A\right|} \\ \Rightarrow A^{-1}\left|A\right|=\mathrm{adj}A \end{gathered}$$

Thus, adj A = $\left|A\right|$ A⁻¹ is correct.

(b) det(A⁻¹) = [det(A)]⁻¹

As we know,
$$\begin{gathered} \left|A^{-1}\right|=\frac{1}{\left|A\right|} \\ \Rightarrow \left|A^{-1}\right|={\left|A\right|}^{-1} \end{gathered}$$

Thus, det(A⁻¹) = [det(A)]⁻¹ is correct.

(c) (AB)⁻¹ = B⁻¹A⁻¹

As we know,
By reversal law of inverse
(AB)⁻¹ = B⁻¹A⁻¹

Thus, (AB)⁻¹ = B⁻¹A⁻¹ is correct.

(d) (A + B)⁻¹ = B⁻¹ + A⁻¹

$$\begin{gathered} {\left(A+B\right)}^{-1}=\frac{1}{\left|A+B\right|}\mathrm{adj}(A+B) \\ \ne \frac{1}{\left|A\right|}\mathrm{adj}\left(A\right)+\frac{1}{\left|B\right|}\mathrm{adj}\left(B\right) \\ \ne {\mathrm{A}}^{-1}+{\mathrm{B}}^{-1} \end{gathered}$$

Thus, (A + B)⁻¹ = B⁻¹ + A⁻¹ is incorrect.


Hence, the correct option is (d).