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NDA Math Determinants

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Question : 58 of 68

Marks: +1, -0

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

1. det (A + B) = det A + det B

2.(A+B)^{-1}=A^{-1}+B^{-1}

Which of the above is/are correct?

1 only
2 only
Both 1 and 2
Neither 1 nor 2

Solution:

Consider A and B to be identity matrix of order 3. Therefore

A=B=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}

\therefore A^{-1}=B^{-1}=\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}

det A = det B = 1

\det A^{-1}=\det B^{-1}=1

det A + det B = 2

\det A^{-1}=\det B^{-1}=2

A+B=2I

=\begin{pmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{pmatrix}

\det(A+B)=2^3

=8

≠ det A + det B

(A+B)^{-1}=0.5I

=\begin{pmatrix}0.5 & 0 & 0 \\ 0 & 0.5 & 0 \\ 0 & 0 & 0.5\end{pmatrix}

A^{-1}+B^{-1}

=2I

=\begin{pmatrix}2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2\end{pmatrix}

\therefore (A+B)^{-1}\neq A^{-1}+B^{-1}

Also

\det(A+B)^{-1}=(0.5)^3

=0.125

\neq \det A^{-1}+\det B^{-1}

Hence none of the above statements are true.

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