Question

Construct a 3×4 matrix, whose elements are given by 
 I. \(a_{ij}=\frac{1}{2}\mid -3i+j\mid\) 
 II. \(a_{ij}=2i-j\)

Answer 1

In general, a 3 × 4 matrix is given by A=\(\begin{bmatrix}        a_{11} & a_{12} & a_{13} &a_{14}           \\[0.3em]        a_{21} & a_{22}           & a_{23}&a_{24} \\[0.3em]        a_{31}           & a_{32} & a_{33} &a_{34}     \end{bmatrix}\)
 

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(i) \(a_{ij}=\mid-3i+j\mid,i=1,2,3\,and\,j=1,2,3,4\)
Therefore \(a_{11}=\frac{1}{2}|-3\times1+1|=\frac{1}{2}|-3+1|=\frac{1}{2}|-2|=\frac{2}{2}=1\)
\(a_{21}=\frac{1}{2}|-3 \times 2+1|=\frac{1}{2}|-6+1|=\frac{1}{2}|-5|=\frac{5}{2}\)
\(a_{31}=\frac{1}{2}|-3\times3+1|=\frac{1}{2}|-9+1|=\frac{1}{2}|-8|=\frac{8}{2}=4\)

\(a_{12}=\frac{1}{2}|-3\times1+2|=\frac{1}{2}|-3+2|=\frac{1}{2}|-1|=\frac{1}{2}\)
\(a_{22}=\frac{1}{2}|-3\times2+2|=\frac{1}{2}|-6+2|=\frac{1}{2}|-4|=\frac{4}{2}=2\)
\(a_{32}=\frac{1}{2}|-3\times3+2|=\frac{1}{2}|-9+2|=\frac{1}{2}|-7|=\frac{7}{2}\)

\(a_{13}=\frac{1}{2}|-3\times1+3|=\frac{1}{2}|-3+3|=0\)
\(a_{23}=\frac{1}{2}|-3\times2+3|=\frac{1}{2}|-6+3|=\frac{1}{2}|-3|=\frac{3}{2}\)
\(a_{33}=\frac{1}{2}|-3\times3+3|=\frac{1}{2}|-9+3|=\frac{1}{2}|-6|=\frac{6}{2}=3\)

\(a_{14}=\frac{1}{2}|-3\times1+4|=\frac{1}{2}|-3+4|=\frac{1}{2}|1|=\frac{1}{2}\)
\(a_{24}=\frac{1}{2}|-3\times2+4|=\frac{1}{2}|-6+4|=\frac{1}{2}|-2|=\frac{2}{2}=1\)
\(a_{34}=\frac{1}{2}|-3\times3+4|=\frac{1}{2}|-9+4|=\frac{1}{2}|-5|=\frac{5}{2}\)

Therefore, the required matrix 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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(ii)\(a_{ij}\)=2i-j  i=1,2,3 and j=1,2,3,4
therefore 
\(a_{11}=2\times1-1=2-1=1\)
\(a_{21}=2\times2-1=4-1=3\)
\(a_{31}=2\times3-1=6-1=5\)

\(a_{12}=2\times1-2=2-2=0\)
\(a_{22}=2\times2-2=4-2=2\)
\(a_{32}=2\times3-2=6-2=4\)

\(a_{13}=2\times1-3=2-3=-1\)
\(a_{23}=2\times2-3=4-3=1\)
\(a_{33}=2\times3-3=6-3=3\)

\(a_{14}=2\times1-4=2-4=-2\)
\(a_{24}=2\times2-4=4-4=0\)
\(a_{34}=2\times3-4=6-4=2\)
Therefore, the required matrix is \(A=\begin{bmatrix}1& 0& -1& -2\\  3& 2& 1& 0\\  5 &4& 3 &2\end{bmatrix}\)