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Study Stuff for MCA Examinations - NIMCET
Study Stuff for MCA Examinations - NIMCET
Previous Year Question (PYQs)
Let A and B be two square matrices of same order satisfying $A^2+5A+5I =0$ and $B^2+3B+I=0$ repectively. Where I is the identity matrix. Then the inverse of the matrix $C= BA+2B+2A+4I$ is
Solution
Given: $A^2 + 5A + 5I = 0 \quad\Rightarrow\quad A^2 = -5A - 5I$$B^2 + 3B + I = 0 \quad\Rightarrow\quad B^2 = -3B - I$
Matrix:
$C = BA + 2B + 2A + 4I$
We want $C^{-1}$.
Step 1: Try a linear expression as inverse
Assume:
$C^{-1} = \alpha AB + \beta A + \gamma B + \delta I$
Test using the fact that quadratic equations give inverses as linear polynomials.
Check option (2):
$AB + A + 3B + 3I$
Multiply with $C$:
$C(AB + A + 3B + 3I)$
$= (BA + 2B + 2A + 4I)(AB + A + 3B + 3I)$
Using reductions:
$A^2 = -5A - 5I$
$B^2 = -3B - I$
Everything simplifies to: $I$
Hence:
$C^{-1} = AB + A + 3B + 3I$
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