Question

If A `= [(1,3),(3,4)]` and A² − kA − 5I = 0, then the value of k is ______.

विकल्प

• 5

• 3

• 7

• 9

Answer 1

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