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3 and 4 .Determinants and Matrices
normal
Let the matrix
$A = \left[ {\begin{array}{*{20}{c}}
{{{10}^{30}} + 5}&{{{10}^{20}} + 4}&{{{10}^{20}} + 6}\\
{{{10}^4} + 2}&{{{10}^8} + 7}&{{{10}^{10}} + 2n}\\
{{{10}^4} + 8}&{{{10}^6} + 4}&{{{10}^{15}} + 9}
\end{array}} \right]$ ,
$n \in N$, then
A
$A$ is invertible for all $n \in N$
B
$A$ is not invertible for all $n \in N$
C
$A$ may or may not be invertible depending on value of $n \in N$
D
Data insufficient
Solution
Replacing even numbers by zero and odd numbers by one, we have
$|A|=\left|\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {0} & {0} & {1}\end{array}\right|=1$
which is an odd number and hence $|\mathrm{A}|$ can not be zero. Hence $A$ is invertible for all $n \in N.$
Standard 12
Mathematics
