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प्रश्न
If A `= [(1,3),(3,4)]` and A2 − kA − 5I = 0, then the value of k is ______.
पर्याय
5
3
7
9
उत्तर
If A `= [(1,3),(3,4)]` and A2 - KA - 5I = 0, then K = 5.
Explanation:
Given,
`A = [(1,3),(3,4)]`
A2 = A.A
`=[(1,3),(3,4)] [(1,3),(3,4)]`
`= [((1)(1)+(3)(3),(1)(3)+(3)( 4)),((3)(1) + ( 4)(3),(3)(3) + ( 4)( 4))]`
`= [(1+9,3+12),(3+12,9+16)]`
`= [(10,15),(15,25)]`
Now, A2 − KA − 5I = 0 [Given]
`[(10,15),(15,25)]-k[(1,3),(3,4)]-5[(1,0),(0,1)]=[(0,0),(0,0)]`
`[(10-k-5,15-3k-0),(15-3k-0,25-4k-5)]=[(0,0),(0,0)]`
`[(5-k,15-3k),(15-3k,20-4k)]=[(0,0),(0,0)]`
On comparing
5 − k = 0
∴ k = 5
OR 15 - 3k = 0
∴ k = 5
OR 20 - 4k = 0
Hence, k = 5
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