.nf
Knuth, vol 2, page 317:
.sp
.EQ 
Q sub n (x sub 1 ,x sub 2 , ... ,x sub n ) ~=~ left {
matrix {
lcol { 1, above x sub 1 , above 
x sub 1 Q sub n-1 (x sub 2 , ... ,x sub n ) +
Q sub n-2 (x sub 3 , ... ,x sub n ) }
lcol {  if above  if above  if }
lcol { n=0; above n=1; above n>1. }
}
.EN
.sp
Knuth, vol 2, p426:
.sp
.ti 3
When zero occurs, the determinant is even easier to compute;
for example, if
.EQ
x sub 11 =0 
.EN
but
.EQ
x sub 21  != 0,
.EN
we have
.sp
.nf
.EQ
det ~
left [ 
matrix {
col { 0 above x sub 21 above x sub 31 above ...  above x sub n1 }
col { x sub 12 above x sub 22 above x sub 32 above ~ above x sub n2 }
col { ...  above ...  above ...  above ~ above ...  }
col { x sub 1n above x sub 2n above x sub 3n above ~ above x sub nn }
}
right ]
.EN
.sp
.ti 5
.EQ
=~-x sub 21 det
left [
matrix {
col { 
x sub 12 above x sub 32 -(x sub 31 /x sub 21 )x sub 22
above ...  above x sub n2 -(x sub n1 /x sub 21 )x sub 22 }
col { ...  above ...  above ~ above ...  }
col {
x sub 1n above x sub 3n -(x sub 31 /x sub 21 )x sub 2n
above ...  above x sub nn -(x sub n1 /x sub 21 )x sub 2n
   }
}
right ]
~~.~~~(8)
.EN

.nf
From John Butcher:
.EQ
A~=~ left [
matrix {
 ccol { a sub 11 above 0 above 0 above 0 above cdot above cdot above cdot above 0 above a sub ni }
 ccol { a sub 12 above a sub 22 above a sub 32 above 0  above cdot above cdot above cdot  above 0 above 0 }
 ccol { 0 above a sub 23 above a sub 33 above a sub 43 above cdot above cdot above cdot above 0 above 0 }
 ccol { 0 above 0 above a sub 34 above a sub 44 above cdot above cdot above cdot above 0 above 0 }
 ccol { cdot above cdot above cdot above cdot above nothing above nothing above nothing above cdot above cdot  }
 ccol { cdot above cdot above cdot above cdot above nothing above nothing above nothing above cdot above cdot  }
 ccol { cdot above cdot above cdot above cdot above nothing above nothing above nothing above cdot above cdot  }
 ccol {0 above 0 above 0 above 0 above cdot above cdot above cdot above a sub n-1,n-1 above a sub n,n-1 }
 ccol {0 above 0 above 0 above 0 above cdot above cdot above cdot above a sub n-1,n above a sub nn }
}
right ]~=
.EN
.nf

.EQ
X~=~ left [
lpile { A above B above C above D above E above F above G above H} ~~~
lpile { I above J above K above L above M above N above O above P} ~~~
lpile { Q above R above S above T above U above V above X above Y} 
right ] ~ finis
.EN
