Construct a 3 2 matrix whose elements are given by a_ i j =1 2 |i…

Question

Construct a 3 $\times$ 2 matrix whose elements are given by $a_{i j}=\frac{1}{2}|i-3 j|$

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Answer

In general a 3 $\times$ 2 matrix is given by $A=\left[\begin{array}{ll} {a_{11}} & {a_{12}} \\ {a_{21}} & {a_{22}} \\ {a_{31}} & {a_{32}} \end{array}\right]$
Now $a_{i j}=\frac{1}{2}$ |i - 3 j|, i = 1, 2, 3 and j = 1, 2.
$\therefore$ $a_{11}=\frac{1}{2}|1-3 \times 1|=1$, $a_{12}=\frac{1}{2}|1-3 \times 2|=\frac{5}{2}$
$a_{21}=\frac{1}{2}|2-3 \times 1|=\frac{1}{2}$, $a_{22}=\frac{1}{2}|2-3 \times 2|=2$
$a_{31}=\frac{1}{2}|3-3 \times 1|=0$, $a_{32}=\frac{1}{2}|3-3 \times 2|=\frac{3}{2}$
Hence the required matrix is given by A = $\left[\begin{array}{cc} {1} & \frac{5}{2} \\ \frac{1}{2} & {2}\\ {0} & {\frac{3}{2}} \end{array}\right]$

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