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Question
Construct an m × n matrix A = [aij], where aij is given by
aij = `|3"i" - 4"j"|/4` with m = 3, n = 4
Solution
To construct a 3 × 4 matrices.
A = `["a"_"ij"]_(5 xx 4) = [("a"_11, "a"_12, "a"_13, "a"_14),("a"_21, "a"_22, "a"_24, "a"_24),("a"_31, "a"_32, "a"_33, "a"_34)]`
a11 = `|3 xx 1 - 4 xx 1|/4`
= `|3 - 4|/4`
= `|- 1|/4`
= `1/4`
a12 = `|3 xx 1 - 4 xx 2|/4`
= `|3 - 8|/4`
= `|- 5|/4`
= `5/4`
a13 = `|3 xx 1 - 4 xx 3|/4`
= `|3 - 12|/4`
= `|- 9|/4`
= `9/4`
a14 = `|3 xx 1 - 4 xx 4|/4`
= `|3 - 16|/4`
= `|- 13|/4`
= `13/4`
a21 = `|3 xx 2 - 4 xx 1|/4`
= `|6 - 4|/4`
= `|2|/4`
= `2/4`
= `1/2`
a22 = `|3 xx 2 - 4 xx 2|/4`
= `|6 - 8|/4`
= `|- 2|/4`
= `2/4`
= `1/2`
a23 = `|3 xx 2 - 4 xx 3|/4`
= `|6 - 12|/4`
= `|- 6|/4`
= `6/4`
= `3/2`
a24 = `|3 xx 2 - 4 xx 4|/4`
= `|6 - 16|/4`
= `|- 10|/4`
= `10/4`
= `5/2`
a31 = `|3 xx 3 - 4 xx 1|/4`
= `|9 - 4|/4`
= `|5|/4`
= `5/4`
a32 = `|3 xx 3 - 4 xx 2|/4`
= `|9 - 8|/4`
= `|1|/4`
= `1/4`
a33 = `|3 xx 3 - 4 xx 3|/4`
= `|9 - 12|/4`
= `|- 3|/4`
= `3/4`
a34 = `|3 xx 3 - 4 xx 4|/4`
= `|9 - 16|/4`
= `|- 7|/4`
= `7/4`
∴ The required 3 × 4 matrix is
A = `[(1/4, 5/4, 9/4, 13/4),(1/2, 1/2, 3/2, 5/2),(5/4, 1/4, 3/4, 7/4)]`
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