SOLVING DETERMINANTS WITHOUT EXPANDING

Question 1 :

Prove that

Solution :

First let us subtract row 2 from row 1

The value of sec²A - tan²A is 1 and the value of tan²A - sec²A  is -1.So, we get

Here column 1 and 3 are identical, so the answer is 0.

Hence proved.

Question 2 :

Show that 

Solution :

First let us subtract row 2 from row 1

After factor out "a" from 1^st row, row 1 and row 3 will be equal.

So the answer is 0.

Hence proved.

Question 3 :

Write the general form of a 3 × 3 skew-symmetric matrix and prove that its determinant is 0.

Solution :

By expanding this, we get

  =  -a(0 + bc) + b(ac - 0)

  =  -a(bc) + b(ac)

  =  -abc + abc

  =  0

Hence determinant of 3 x 3 skew matrix is 0.

Question 4 :

If

prove that a, b, c are in G.P. or a is a root of ax² + 2bx + c = 0.

Solution :

Hence a,b and c are in G.P

Question 5 :

prove that

Solution :

By using the property let us write the given determinant as sum of two determinants. 

Now let us factor out a,b and c from 1^st, 2^nd and 3^rd row respectively.

By interchanging R₁ and R₃, we get

  =   0

Hence proved.

Question 6 :

If a, b, c are p^th, q^th and r^th terms of an A.P, find the value of

Solution :

General term of A.P

a_n  =  a + (n - 1)d

Now we have to replace a, b and c by (1), (2) and (3) respectively.

In the first determinant, 1^st and 3^rd rows are identical.

In the second determinant, 1^st and 2^nd rows are identical.

In the third determinant, 1^st and 3^rd rows are identical.

  =  a(0) + d(0) - 0

  =  0

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