Engineering mathematics

Question

if the sum of the diagonal elements of a 2x2 matrix is (-6) then the maximum possible value of determinant of the matrix is

Comments

$A = \begin{bmatrix} a & b\\ c& d \end{bmatrix}$

$\left | A \right | = ad - cb$

 

For max $cb$ must be $0$

$\left | A \right | = ad$

$a + d = -6$

 

$a$ and $d$ cannot be +tive as the sum will not be -6

 

if $a$ is +tive or $d$ is -tive or vice versa then it will not give the maximum value [ product will be not maximum]

Possible values of $a$ and $d$

$a$	$d$	$a+d$ / $a.d$
$0$	$-6$	$-6 / 0$
$-1$	$-5$	$-6 / 5$
$-2$	$-4$	$-6 / 8$
$-3 $	$-3$	$-6 / 9$ $\bigstar$

 

Answer 1

 

ANSWER IS +9

Comments

why b*c needs to be 0 always. why it cannot be -ve? for maximum value?

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YA IT CAN BE NEGATIVE BUT A SOLUTION EXIST FOR THIS PROBLEM IF B=C AND FRAME SUCH MATRIX IN THAT CASE DETERMINENT WILL BECOME  AD-B² 

AND B²  WILL ALWAYS BE POSITIVE SO IT HAVE TO BE ZERO ...AND THEN WE CAN EITHER USING CONCEPT OF MAXIMA FINDING USING DIFFERENTIATION OR JUST BY ITERATING AS I DID ...BUT AS FAR AS INFINITY IS CORRECT IF MATRIX IS NOT GIVEN  

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ok ok got it. it says about maximum possible value.

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there is no condition for b=c

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@Deepanshu   @ arvin  QUESTION IS INCOMPLETE ORIGINAL CONTAINS "A IS SYMMETRIC MATRIX"......THEN ONLY WE CAN FIND MAXIMUM OTHERWISE NO WAY TO FIND MAXIMUM.

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@eyeamj this was what i was thinking but as it says possible value i thought it was with respect to sum of diagonal.

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AND  I WAS  JUST TRYING TO FIND AN ANSWER WHICh WAS WRONG APPROACH ....

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@eyeamj : yes and i too considered because possible value can be 9 or 10 or anything as nothing is mentioned..

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yep some more conditions are required in this question

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Thank you guys for the explanation. I was not considering bc =0. Thats why making the mistake.

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What if bc is negative

Answer 2

Suppose i put b=18 and c=-2

and a and d according to answers below  -3

so determinant is 9-(18*-2)=45

i think this question is wrong we can find minimum not maximum

Comments

@Deepanshu: it is asking maximum possible value.

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i dont get it