HCU MCA 2013 Paper

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Question : 51 of 75

Marks: +1, -0

Suppose a, b and c are all > 0, and if D is the determinant below, which of the following is true about D ?
D =

|\table a,b,c;b,c,a;c,a,b|

\I

. D > 0

\II

. D < 0

\III

.D = 0

Only $\I$
Both $\I$ and $\II$
Only $\II$
Both $\II$ and $\III$

Solution:

(a) Given that, a, b,c > 0
and

\;\;\;\;\;\;\;\;

D =

|\table a,b,c;b,c,a;c,a,b|

Using operation

\R_2→\C_1 + \C_2 + \C_3,

\;\;\;\;\;\;\;\;\;\;\;\;\;\;

D =

|\table a+b+c,b,c;a+b+c,c,a;a+b+c,a,b|

\;\;\;\;\;\;\;\;\;\;\;\;\;\;

D =

(a+b+c)|\table 1,b,c;1,c,a;1,a,b|

Using operation

\R_2→\R_2-\R_1, \R_3→\R_3-\R_1

\;\;\;\;\;\;\;\;\;\;\;\;\;\;

D =

(a+b+c)|\table 1,b,c;0,c-b,a-c;0,a-b,b-c|

Expanding along

\C_1

D =

(a + b + c) \{(c - b) (b - c) - (a - c) (a - b)\}

\;= (a + b + c) \{bc - b^2 - c ^2 + bc - a^2 + ac + ba - bc\}

\;=-1/2(a + b + c) \{(a - b)^2 + (b - c)^2 + (c - a)^2\}

Since, a ,b ,c > 0
Then, D > 0

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