Question

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

`a_(ij) = 1/2 |-3i + j|`

Answer 1

In general, a 3 × 4 matrix is given by A = `[(a_11, a_12, a_13, a_14), (a_21, a_22, a_23, a_24), (a_31, a_32, a_33, a_34)]`

`a_(ij) = 1/2 |-3i + j|, i = 1, 2, 3 and j = 1, 2, 3, 4`

∴ `a_11 = 1/2 |-3 xx 1 + 1|`

= `1/2 |-3 + 1|`

= `1/2 |-2|`

= `2/2`

= 1

`a_21 = 1/2 |-3 xx 2 + 1|`

= `1/2 |-6 + 1|`

=`1/2 |-5|`

= `5/2`

`a_31 = 1/2 |-3 xx 3 + 1|`

= `1/2| -9 + 1|`

= `1/2 |-8|`

= `8/2`

= 4

`a_12 = 1/2 |-3 xx 1 + 2|`

= `1/2 |-3 + 2|`

= `1/2 |-1|`

= 1

`a_22 = 1/2 |-3 xx 2 + 2|`

= `1/2 |-6 + 2|`

= `1/2 |-4|`

= `4/2`

= 2

`a_32 = 1/2 |-3 xx 3 + 2|`

= `1/2 |-9 + 2|`

= `1/2 |-7|`

= `7/2`

`a_13 = 1/2 |-3 xx 1 + 3|`

= `1/2 |-3 + 3|`

= 0

`a_23 = 1/2 |-3 xx 2 + 3|`

= `1/2 |-6 + 3|`

= `1/2 |-3|`

= `3/2`

`a_33 = 1/2 |-3 xx 3 + 3|`

= `1/2 |-9 + 3|`

= `1/2 |-6|`

= `6/2`

= 3

`a_14 = 1/2 |-3 xx 1 + 4|`

= `1/2 |-3 + 4|`

= `1/2 |1|`

= `1/2`

`a_24 = 1/2 |-3 xx 2 + 4|`

= `1/2 |-6 + 4|` 

= `|-2|/2`

= `2/2`

= 1

`a_34 = 1/2 |-3 xx 3 + 4|`

= `1/2 |-9 + 4|`

= `1/2 |-5|`

= `5/2`

Therefore, the required matrix is A = `[(1, 1/2, 0, 1/2), (5/2, 2, 3/2, 1), (4, 7/2, 3, 5/2)]`.