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Question

Construct a 3 × 4 matrix A = [aij] whose elements aij are given by:
(i) aij = i + j
(ii) aij = i − j
(iii) aij = 2i
(iv) aij = j
(v) aij = 12 -3i+j

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Solution

i

aij=i+j

Here,
a11=1+1=2 , a12=1+2=3 , a13=1+3=4 , a14=1+4=5a21=2+1=3 , a22=2+2=4 , a23=2+3=5 , a24=2+4=6a31=3+1=4 , a32=3+2=5 , a33=3+3=6 and a34=3+4=7

So, the required matrix is 234534564567.

ii

aij=i-j

Here,
a11=1-1=0 , a12=1-2=-1 , a13=1-3=-2 , a14=1-4=-3a21=2-1=1 , a22=2-2=0 , a23=2-3=-1 , a24=2-4=-2a31=3-1=2 , a32=3-2=1 , a33=3-3=0 and a34=3-4=-1

So, the required matrix is 0-1-2-310-1-2210-1.

iii

aij=2i

Here,
a11=21=2 , a12=21=2 , a13=21=2 , a14=21=2a21=22=4 , a22=22=4 , a23=22=4 , a24=22=4a31=23=6 , a32=23=6 , a33=23=6 and a34=23=6

So, the required matrix is 222244446666.

iv

aij=j

Here,
a11=1 , a12=2 , a13=3 , a14=4a21=1 , a22=2 , a23=3 , a24=4a31=1 , a32=2 , a33=3 and a34=4

So, the required matrix is 123412341234.

v

aij=12-3i+j

Here,

a11=12-31+1=12-2=1 , a12=12-31+2=12-1=12 , a13=12-31+3=120=02=0 , a14=12-31+4=121=12a21=12-32+1=12-5=52 , a22=12-32+2=12-4=2 , a23=12-32+3=12-3=32 , a24=12-32+4=12-2=1a31=12-33+1=12-8=4 , a32=12-33+2=12-7=72 , a33=12-33+3=12-6=3 and a34=12-33+4=12-5=52


So, the required matrix is 112012522321472352.

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