Q. 324.3( 3 Votes )
Show that the matrix A = 
satisfies the equation π³2 + 4π³ β 42 = 0 and hence find A-1.
Answer :
Given: 
To show: Matrix A satisfies the equation x2 + 4x β 42 = 0
If Matrix A satisfies the given equation then
A2 + 4A β 42 = 0
Firstly, we find the A2
Taking LHS of the given equation .i.e.
A2 + 4A β 42
= O
= RHS
β΄ LHS = RHS
Hence matrix A satisfies the given equation x2 + 4x β 42 = 0
Now, we have to find A-1
Finding A-1 using given equation
A2 + 4A β 42 = O
Post multiplying by A-1 both sides, we get
(A2 + 4A β 42)A-1 = OA-1
β A2.A-1 + 4A.A-1 β 42.A-1 = O [OA-1 = O]
β A.(AA-1) + 4I β 42A-1 = O [AA-1 = I]
β A(I) + 4I β 42A-1 = O
β A + 4I β 42A-1 = O
β A + 4I β O = 42A-1
Ans. 
.
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