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GO Classes DPP | GATE CS, DA | Linear Algebra | Matrix

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10 10 votes

Let $A$ be an $n\times n$ invertible matrix with real entries whose row sums are all equal to $c$. Consider the following statements:

  1. Every row in the matrix $2A$ sums to $2c$.
     
  2. Every row in the matrix $A^{2}$ sums to $c^{2}$.
     
  3. Every row in the matrix $A^{-1}$ sums to $c^{-1}$.
     

Which of the following is TRUE?

  1. none of the statements $(1), (2), (3)$ is correct
     
  2. statement $(1)$ is correct but not necessarily statements $(2)$ or $(3)$
     
  3. statement $(1)$  and $(2)$ are correct but not necessarily statement $(3)$
     
  4. all the three statements $(1), (2),$ and $(3)$ are correct

3 Answers

4 4 votes
$$
A\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)
$$
gives a column vector each of whose entries is the row sum, $r$ say.

Then
$$
\begin{gathered}
2 A\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)=2\left(\begin{array}{c}
r \\
r \\
\ldots \\
r
\end{array}\right)=\left(\begin{array}{c}
2 r \\
2 r \\
\ldots \\
2 r
\end{array}\right) . \\\\
A^2\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)=A\left(\begin{array}{c}
r \\
r \\
\ldots \\
r
\end{array}\right)=r A\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)=\left(\begin{array}{c}
r^2 \\
r^2 \\
\ldots \\
r^2
\end{array}\right) \\\\
A^{-1}\left(\begin{array}{c}
r \\
r \\
\ldots \\
r
\end{array}\right)=A^{-1} A\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)=\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)
\end{gathered}
$$
and therefore
$$
A^{-1}\left(\begin{array}{c}
1 \\
1 \\
\ldots \\
1
\end{array}\right)=\left(\begin{array}{c}
r^{-1} \\
r^{-1} \\
\ldots \\
r^{-1}
\end{array}\right)
$$
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