1. Find LQ Decomposition ...
`[[1,-1,4],[1,4,-2],[1,4,2],[1,-1,0]]`

Solution:

Here `A`

=

`1` `-1` `4` `1` `4` `-2` `1` `4` `2` `1` `-1` `0`

`A`

=

`1` `-1` `4` `1` `4` `-2` `1` `4` `2` `1` `-1` `0`

`A'`

=

`1` `1` `1` `1` `-1` `4` `4` `-1` `4` `-2` `2` `0`

Now, UR Decomposition of `A'` by GramSchmidt Method

`q_1'`

`=a_1`

=

`1` `-1` `4`

=

`1` `-1` `4`

`r_(11)=||q_1'||=sqrt((1)^2+(-1)^2+(4)^2)=sqrt(18)=4.2426406871`

`q_1 = 1/(||q_1'||) * q_1'`

=

`1/4.2426406871 * `

`1` `-1` `4`

=

`0.2357022604` `-0.2357022604` `0.9428090416`

`r_(12)=q_1^T * a_2`

=

[ `0.2357022604` `-0.2357022604` `0.9428090416` ]

`xx`

`1` `4` `-2`

`=-2.5927248644`

`q_2'`

`=a_2-r_(12) * q_1`

=

`1` `4` `-2`  `+2.5927248644`  `0.2357022604` `-0.2357022604` `0.9428090416`

=

`1.6111111111` `3.3888888889` `0.4444444444`

`r_(22)=||q_2'||=sqrt((1.6111111111)^2+(3.3888888889)^2+(0.4444444444)^2)=sqrt(14.2777777778)=3.7785946829`

`q_2 = 1/(||q_2'||) * q_2'`

=

`1/3.7785946829 * `

`1.6111111111` `3.3888888889` `0.4444444444`

=

`0.4263783883` `0.8968648858` `0.1176216244`

`r_(13)=q_1^T * a_3`

=

[ `0.2357022604` `-0.2357022604` `0.9428090416` ]

`xx`

`1` `4` `2`

`=1.178511302`

`r_(23)=q_2^T * a_3`

=

[ `0.4263783883` `0.8968648858` `0.1176216244` ]

`xx`

`1` `4` `2`

`=4.2490811804`

`q_3'`

`=a_3-r_(13) * q_1-r_(23) * q_2`

=

`1` `4` `2`  `-1.178511302`  `0.2357022604` `-0.2357022604` `0.9428090416`  `-4.2490811804`  `0.4263783883` `0.8968648858` `0.1176216244`

=

`-1.0894941634` `0.46692607` `0.3891050584`

`r_(33)=||q_3'||=sqrt((-1.0894941634)^2+(0.46692607)^2+(0.3891050584)^2)=sqrt(1.5564202335)=1.2475657231`

`q_3 = 1/(||q_3'||) * q_3'`

=

`1/1.2475657231 * `

`-1.0894941634` `0.46692607` `0.3891050584`

=

`-0.8732960062` `0.3742697169` `0.3118914308`

`r_(14)=q_1^T * a_4`

=

[ `0.2357022604` `-0.2357022604` `0.9428090416` ]

`xx`

`1` `-1` `0`

`=0.4714045208`

`r_(24)=q_2^T * a_4`

=

[ `0.4263783883` `0.8968648858` `0.1176216244` ]

`xx`

`1` `-1` `0`

`=-0.4704864975`

`r_(34)=q_3^T * a_4`

=

[ `-0.8732960062` `0.3742697169` `0.3118914308` ]

`xx`

`1` `-1` `0`

`=-1.2475657231`

`q_4'`

`=a_4-r_(14) * q_1-r_(24) * q_2-r_(34) * q_3`

=

`1` `-1` `0`  `-0.4714045208`  `0.2357022604` `-0.2357022604` `0.9428090416`  `+0.4704864975`  `0.4263783883` `0.8968648858` `0.1176216244`  `+1.2475657231`  `-0.8732960062` `0.3742697169` `0.3118914308`

=

`0` `0` `0`

`r_(44)=||q_4'||=sqrt((0)^2+(0)^2+(0)^2)=sqrt(0)=0`

`q_4 = 1/(||q_4'||) * q_4'`

=

`1/0 * `

`0` `0` `0`

=

`-0.5494422558` `-0.1373605639` `-0.8241633837`

`U`

`=[q_1,q_2,q_3,q_4]`

=

`0.2357022604` `0.4263783883` `-0.8732960062` `-0.5494422558` `-0.2357022604` `0.8968648858` `0.3742697169` `-0.1373605639` `0.9428090416` `0.1176216244` `0.3118914308` `-0.8241633837`

`R`

=

`r_(11)` `r_(12)` `r_(13)` `r_(14)` `0` `r_(22)` `r_(23)` `r_(24)` `0` `0` `r_(33)` `r_(34)` `0` `0` `0` `r_(44)`

=

`4.2426406871` `-2.5927248644` `1.178511302` `0.4714045208` `0` `3.7785946829` `4.2490811804` `-0.4704864975` `0` `0` `1.2475657231` `-1.2475657231` `0` `0` `0` `0`

---

Now, L and Q from R and U

`L=R^T`

=

`4.2426406871` `0` `0` `0` `-2.5927248644` `3.7785946829` `0` `0` `1.178511302` `4.2490811804` `1.2475657231` `0` `0.4714045208` `-0.4704864975` `-1.2475657231` `0`

`Q=U^T`

=

`0.2357022604` `-0.2357022604` `0.9428090416` `0.4263783883` `0.8968648858` `0.1176216244` `-0.8732960062` `0.3742697169` `0.3118914308` `-0.5494422558` `-0.1373605639` `-0.8241633837`

---

checking `L xx Q = A?`

`L xx Q`

=

`4.2426406871` `0` `0` `0` `-2.5927248644` `3.7785946829` `0` `0` `1.178511302` `4.2490811804` `1.2475657231` `0` `0.4714045208` `-0.4704864975` `-1.2475657231` `0`

`xx`

`0.2357022604` `-0.2357022604` `0.9428090416` `0.4263783883` `0.8968648858` `0.1176216244` `-0.8732960062` `0.3742697169` `0.3118914308` `-0.5494422558` `-0.1373605639` `-0.8241633837`

=

`1` `-1` `4` `1` `4` `-2` `1` `4` `2` `1` `-1` `0`

and `A`

=

`1` `-1` `4` `1` `4` `-2` `1` `4` `2` `1` `-1` `0`

Solution is possible.