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Question
If X = `|(1 , -2),(1 , 3)|` , Y = `|(-3 , 0),(4 , 1)|` and Z = `|(5 , -1),(3 , 2)|` , prove that X (Y + Z) = XY + XZ
Solution
X = `|(1 , -2),(1 , 3)|_(2 xx 2)` Y = `|(-3 , 0),(4 , 1)|_(2 xx 2)` Z = `|(5 , -1),(3 , 2)|_(2 xx 2)`
(Y + Z) = `|(-3 , 0),(4 , 1)| + |(5 , -1),(3 , 2)| = |(2 , -1),(7 , 3)|_(2 xx 2)`
X (Y + Z) = `|(1 , -2),(1 , 3)| |(2 , -1),(7 , 3)|`
`= |(2 - 14 , -1-6),(2 + 21 , -1+9)|`
X (Y + Z) = `|(-12 , -7),(23 , 8)|_(2 xx 2)` ........(1)
XY = `|(1 , -2),(1 , 3)| |(-3 , 0),(4 , 1)|`
=`|(-3-8 , 0-2),(-3+12 , 0 + 3)|`
`= |(-11 , -2),(9 , 3)|_(2 xx 2)`
XZ = `|(1 , -2),(1 , 3)| |(5 , -1),(3 , 2)|`
`= |(5 - 6 , -1-4 ),(5 + 9 , -1 + 6)|`
`= |(-1 , -5),(14 , 5)|_(2 xx 2)`
XY + XZ = `|(-11 , -2),(9 , 3)| + |(-1 , -5),(14 , 5)| = |(-12 , -7),(23 , 8)|_(2 xx 2)` ......(2)
from (1) and (2) X(Y + Z) = XY + YZ
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