Q5.    Construct a 3 × 4 matrix, whose elements are given by:

         (i)   a_{ij} = \frac{1}{2}|-3i + j|

Answers (1)

 

 (i) 

 a_{ij} = \frac{1}{2}|-3i + j|

      a_1_1 = \frac{\left | -3+1 \right |}{2}=\frac{2}{2}=1                 a_1_2 = \frac{\left | (-3\times 1)+2 \right |}{2}=\frac{1}{2}                      a_1_3 = \frac{\left | (-3\times 1)+3 \right |}{2}=0                            

      a_2_1 = \frac{\left | (-3\times 2)+1 \right |}{2}=\frac{5}{2}                a_2_2 = \frac{\left | (-3\times 2)+2 \right |}{2}=\frac{4}{2}=2        a_2_3 = \frac{\left | (-3\times 2)+3 \right |}{2}=\frac{\left | -6+3 \right |}{2}=\frac{\left | -3 \right |}{2} =\frac{3}{2}        

 a_3_1 = \frac{\left | (-3\times 3)+1 \right |}{2}=\frac{8}{2}=4          a_3_2 = \frac{\left | (-3\times 3)+2 \right |}{2}=\frac{7}{2}                 a_3_3 = \frac{\left | (-3\times 3)+3 \right |}{2}=\frac{\left | -9+3 \right |}{2}=\frac{\left | -6 \right |}{2} =\frac{6}{2}=3                      

a_1_4 = \frac{\left | (-3\times 1)+4 \right |}{2}=\frac{\left | -3+4 \right |}{2}=\frac{\left | 1 \right |}{2} =\frac{1}{2}                a_2_4 = \frac{\left | (-3\times 2)+4 \right |}{2}=\frac{\left | -6+4 \right |}{2}=\frac{\left | -2 \right |}{2} =\frac{2}{2}=1

a_3_4 = \frac{\left | (-3\times 3)+4 \right |}{2}=\frac{\left | -9+4 \right |}{2}=\frac{\left | -5 \right |}{2} =\frac{5}{2}

 

Hence, the required matrix of the given order is

  A = \begin{bmatrix} 1& \frac{1}{2} & 0&\frac{1}{2} \\ \frac{5}{2} & 2&\frac{3}{2}&1 \\4&\frac{7}{2}&3&\frac{5}{2}\end{bmatrix}

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