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

MCQ

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

A
$A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{3}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{3}{2}}
\end{array}} \right]$
B
$A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{5}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{1}{2}}
\end{array}} \right]$
✓
$A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{5}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{3}{2}}
\end{array}} \right]$
D
$A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{5}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{5}{2}}
\end{array}} \right]$

✓

Answer

Correct option: C.

$A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{5}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{3}{2}}
\end{array}} \right]$

c
In general a $3 \times 2$ matrix is given by $A\, = \,\left[ {\begin{array}{*{20}{c}}
{{a_{11}}}&{{a_{12}}} \\
{{a_{21}}}&{{a_{22}}} \\
{{a_{31}}}&{{a_{32}}}
\end{array}} \right]$

Now ${a_{ij}} = \frac{1}{2}|i - 3j|,$ $i=1,\,2,\,3$ 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 $A\, = \,\left[ {\begin{array}{*{20}{c}}
1&{\frac{5}{2}} \\
{\frac{1}{2}}&2 \\
0&{\frac{3}{2}}
\end{array}} \right]$

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