Example.1
Express D =
as the product of two
determinants and evaluate it.
Solution
D =![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image002.gif)
=
´
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image004.gif)
= 2 (a – b) (b – c)
(c – a) (x – y) (y – z) (z – x).
Example.2
Let a, b, c be real numbers with a2 + b2
+ c2 = 1. Show that the equation .
represents a straight
line.
Solution
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image006.gif)
C1 → aC1
+ bC2 + cC3
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image007.gif)
applying C2 → C2
– bC1 and C3 → C3 – cC1
applying R3 → R3
+ xR1 + yR2, we get
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image010.gif)
&⇒ (x2 + y2 +1)(aby + a2x
+ ac) = 0
&⇒ ax + by + c = 0
Example.3
Prove that ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image011.gif)
Solution
Let D
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image012.gif)
Multiplying C1,
by a Then D = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image013.gif)
Applying C1
→ C1 + bC2
+ cC3, we get
D = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image014.gif)
Taking (a2
+ b2 +c2) common from C1,
\ D
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image015.gif)
Applying C2
→ C2 – bC1
and C3 → C3 - cC1,
Then D = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image016.gif)
Multiplying R1
by x,
D
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image018.gif)
Applying R1
→ R1 + yR2
+ zR3 ,
D
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image019.gif)
Expanding along R1
\
D = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image020.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image021.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image022.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image023.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image024.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image025.gif)
Hence D = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image025.gif)
Example.4
Prove that
=
2 sin
.
Solution
Do C1 → C1+C2+C3,
followed by R2 →
R2 – R1 and R3 → R3 – R1. Then expand along C1,
so that the determinant is.
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image028.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image029.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image030.gif)
=
´ [
]
This can be simplified using cosq + cos2q+cos3q
= 2 cos 2q cosq + cos 2q= cos2q (2cos q+1),
cosq
– cos 2q =
cosq
– cos 3q =
and cos 3q – cos 2q = – 2 sin
.
The determinant can now be written cos 2q (2 cosq +1) ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image036.gif)
= 4 sin2 ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image037.gif)
Further simplification follows after noting
that
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image038.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image039.gif)
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image040.gif)
=
and ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image042.gif)
After substituting these results, the
determinant equals
![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image043.gif)
= ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image044.gif)
Using the identity (a–b) (a2 + ab
+ b2) = a3–b3. This proves the results.
Example.5
If numbers n , r
are two different positive integers such that n≥ r+2 and it is given that D( n, r) =
,
then show that D(n , r) =
D
( n-1, r-1) .
Solution
We know mCk =
m-1Ck-1
Now D (n, r) = ![](http://www.quizsolver.com/radix/dth/notif/Determinanant%20SUB%202_files/image048.gif)
=
D(n –1, r –1)