Find LQ Decomposition ...
`[[1,2,4],[0,0,5],[0,3,6]]`

Solution:

Here `A`

=

`1` `2` `4` `0` `0` `5` `0` `3` `6`

`A`

=

`1` `2` `4` `0` `0` `5` `0` `3` `6`

`A'`

=

`1` `0` `0` `2` `0` `3` `4` `5` `6`

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

`q_1'`

`=a_1`

=

`1` `2` `4`

=

`1` `2` `4`

`r_(11)=||q_1'||=sqrt((1)^2+(2)^2+(4)^2)=sqrt(21)=4.582575695`

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

=

`1/4.582575695 * `

`1` `2` `4`

=

`0.2182178902` `0.4364357805` `0.8728715609`

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

=

[ `0.2182178902` `0.4364357805` `0.8728715609` ]

`xx`

`0` `0` `5`

`=4.3643578047`

`q_2'`

`=a_2-r_(12) * q_1`

=

`0` `0` `5`  `-4.3643578047`  `0.2182178902` `0.4364357805` `0.8728715609`

=

`-0.9523809524` `-1.9047619048` `1.1904761905`

`r_(22)=||q_2'||=sqrt((-0.9523809524)^2+(-1.9047619048)^2+(1.1904761905)^2)=sqrt(5.9523809524)=2.4397501824`

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

=

`1/2.4397501824 * `

`-0.9523809524` `-1.9047619048` `1.1904761905`

=

`-0.3903600292` `-0.7807200584` `0.4879500365`

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

=

[ `0.2182178902` `0.4364357805` `0.8728715609` ]

`xx`

`0` `3` `6`

`=6.5465367071`

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

=

[ `-0.3903600292` `-0.7807200584` `0.4879500365` ]

`xx`

`0` `3` `6`

`=0.5855400438`

`q_3'`

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

=

`0` `3` `6`  `-6.5465367071`  `0.2182178902` `0.4364357805` `0.8728715609`  `-0.5855400438`  `-0.3903600292` `-0.7807200584` `0.4879500365`

=

`-1.2` `0.6` `0`

`r_(33)=||q_3'||=sqrt((-1.2)^2+(0.6)^2+(0)^2)=sqrt(1.8)=1.3416407865`

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

=

`1/1.3416407865 * `

`-1.2` `0.6` `0`

=

`-0.894427191` `0.4472135955` `0`

`U`

`=[q_1,q_2,q_3]`

=

`0.2182178902` `-0.3903600292` `-0.894427191` `0.4364357805` `-0.7807200584` `0.4472135955` `0.8728715609` `0.4879500365` `0`

`R`

=

`r_(11)` `r_(12)` `r_(13)` `0` `r_(22)` `r_(23)` `0` `0` `r_(33)`

=

`4.582575695` `4.3643578047` `6.5465367071` `0` `2.4397501824` `0.5855400438` `0` `0` `1.3416407865`

---

Now, L and Q from R and U

`L=R^T`

=

`4.582575695` `0` `0` `4.3643578047` `2.4397501824` `0` `6.5465367071` `0.5855400438` `1.3416407865`

`Q=U^T`

=

`0.2182178902` `0.4364357805` `0.8728715609` `-0.3903600292` `-0.7807200584` `0.4879500365` `-0.894427191` `0.4472135955` `0`

---

checking `L xx Q = A?`

`L xx Q`

=

`4.582575695` `0` `0` `4.3643578047` `2.4397501824` `0` `6.5465367071` `0.5855400438` `1.3416407865`

`xx`

`0.2182178902` `0.4364357805` `0.8728715609` `-0.3903600292` `-0.7807200584` `0.4879500365` `-0.894427191` `0.4472135955` `0`

=

`1` `2` `4` `0` `0` `5` `0` `3` `6`

and `A`

=

`1` `2` `4` `0` `0` `5` `0` `3` `6`

Solution is possible.

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