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$\mathrm{M}$ is a matrix with real entries given by
$\mathrm{M}=\left[\begin{array}{lll}
4 & \mathrm{k} & 0 \\
6 & 3 & 0 \\
2 & \mathrm{t} & \mathrm{k}
\end{array}\right]$
Which of the following conditions guarantee the invertibility of $\mathrm{M}$ ?
$1. k \neq 2$
$2. \mathrm{k} \neq 0$
$3. \mathrm{t} \neq 0$
$4. \mathrm{t} \neq 1$
Select the correct answer using the code given below:
Options:
$\mathrm{M}=\left[\begin{array}{lll}
4 & \mathrm{k} & 0 \\
6 & 3 & 0 \\
2 & \mathrm{t} & \mathrm{k}
\end{array}\right]$
Which of the following conditions guarantee the invertibility of $\mathrm{M}$ ?
$1. k \neq 2$
$2. \mathrm{k} \neq 0$
$3. \mathrm{t} \neq 0$
$4. \mathrm{t} \neq 1$
Select the correct answer using the code given below:
Solution:
1965 Upvotes
Verified Answer
The correct answer is:
1 and 2
As given $\mathrm{M}=\left[\begin{array}{lll}4 & \mathrm{k} & 0 \\ 6 & 3 & 0 \\ 2 & \mathrm{t} & \mathrm{k}\end{array}\right]$
$\mathrm{M}$ will be invertible, if
$\left[\begin{array}{lll}4 & \mathrm{k} & 0 \\ 6 & 3 & 0 \\ 2 & \mathrm{t} & \mathrm{k}\end{array}\right] \neq 0$
$\Rightarrow \mathrm{k} \neq 0$ or $\mathrm{k}(12-6 \mathrm{k}) \neq 0$
$\Rightarrow \mathrm{k} \neq 0, \mathrm{k} \neq 2$
Thus, statement (1) and (2) are correct.
$\mathrm{M}$ will be invertible, if
$\left[\begin{array}{lll}4 & \mathrm{k} & 0 \\ 6 & 3 & 0 \\ 2 & \mathrm{t} & \mathrm{k}\end{array}\right] \neq 0$
$\Rightarrow \mathrm{k} \neq 0$ or $\mathrm{k}(12-6 \mathrm{k}) \neq 0$
$\Rightarrow \mathrm{k} \neq 0, \mathrm{k} \neq 2$
Thus, statement (1) and (2) are correct.
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