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3 and 4 .Determinants and Matrices
medium
Construct a $2 \times 2$ matrix, $A=\left[a_{ij}\right]$, whose elements are given by : $a_{i j}=\frac{i}{j}$.
Option A
Option B
Option C
Option D
Solution
$(ii)$ Since it is a $2 \times 2$ matrix
it has $2$ rows and $2$ column.
Let matrix be $A$
Where $A=\left[\begin{array}{ll}a_{11} & a_{12} \\ a_{21} & a_{22}\end{array}\right]$
Now it is given that
$a _{ ij }=\frac{ i }{ j }$
| $a_{i j}$ | $i=, j=$ | $a_{i j}=\frac{(i+j)^2}{2}$ |
| $a_{11}$ | $i=1, j=1$ | $a_{11}=\frac{1}{1}=1$ |
| $a_{12}$ | $i=1, j=2$ | $a_{12}=\frac{1}{2}$ |
| $a_{21}$ | $i=2, j=1$ | $a_{21}=\frac{2}{1}=2$ |
| $a_{22}$ | $i=2, j=2$ | $a_{22}=\frac{2}{2}=1$ |
Hence, the required matrix $A$ is
$A\left[\begin{array}{ll}a_-{11} & a_{12} \\a_{21} & a_{22}\end{array}\right]=\left[\begin{array}{ll}1 & \frac{1}{2} \\2 & 1\end{array}\right]$
Standard 12
Mathematics
