Question

Given that A=
5 7 3
1 5 2
3 2 1
and A³ -KA²+lA = I; where I is a 3x3 unit matrix, then the value of k+l is​

Answer 1

Given : $A = \left[\begin{array}{ccc}5&7&3\\1&5&2\\3&2&1\end{array}\right]$

A³ - KA² + LA  = I

To Find : K + L

Solution:

$A = \left[\begin{array}{ccc}5&7&3\\1&5&2\\3&2&1\end{array}\right]$

$A^2 = \left[\begin{array}{ccc}41&76&32\\16&36&15\\20&33&14\end{array}\right]$

$A^3 = \left[\begin{array}{ccc}377&731&307\\161&322&135\\175&333&140\end{array}\right]$

$I = \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]$

377 - K41  + L5  = 1  => 41K - 5L   = 376   => 287K - 35L = 2632

731 - 76K + L7  = 0 =>  76Kk - 7L  =  731  =>  380K - 35L  = 3655

=> 93K = 1023

=> K= 11

=> L= 15

K + L = 11 + 15 = 26

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