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

Answer 1

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