Home
Class 12
MATHS
For the matrix A=[[1, 1, 1], [1, 2, -3],...

For the matrix `A=[[1, 1, 1], [1, 2, -3], [2, -1, 3]]` . Show that `A^3-6A^2+5A+11\ I_3=O` . Hence, find `A^(-1)` .

Text Solution

Verified by Experts

`A^2=[[1, 1, 1], [1, 2, -3], [2, -1, 3]] [[1, 1, 1], [1, 2, -3], [2, -1, 3]]`
`=[[4, 2, 1], [-3, 8, -14], [7, -3, 14]`
`A^3=[[4, 2, 1], [-3, 8, -14], [7, -3, 14]] [[1, 1, 1], [1, 2, -3], [2, -1, 3]] `
`=[[8, 7, 1], [-23, 27, -69], [32, -13, 58]]`
`A^3-6A^2+5A+11I=[[0,0,0],[0,0,0],[0,0,0]]=0`
Multiplying eqn (1) by `A ^(−1)` , we get, `11A^(-1)=-A^2+6A-5I` ...
Promotional Banner

Topper's Solved these Questions

  • ADJOINTS AND INVERSE OF MATRIX

    RD SHARMA|Exercise QUESTION|1 Videos

  • ALGEBRA OF MATRICES

    RD SHARMA|Exercise Solved Examples And Exercises|410 Videos

Similar Questions

Explore conceptually related problems

Find the rank of the matrix [[1,1,1],[1,2,3],[1,3,6]]

For the matrix A=det[[1,1,11,2,-32,-1,3]], then that A^(3)-6A^(2)+5A+4I=0., find A^(-1)

If A= [[3,1] , [-1,2]] then show that A^2 - 5A+7I =0 Hence find A^(-1)

If A=[[2,-1, 1],[-1 ,2,-1],[ 1, 1, 2]] .Verify that A^3-6A^2+9A-4I=0 and hence find A^(-1) .

If A=[{:(3,1),(-1,2):}] , show that A^(2)-5A+7I=O . Hence, find A^(-1) .

Q.3.Find the rank of the matrix [[1,1,-1],[2,-3,4],[3,-2,3]]

Find the Rank of the matrix: [[1,1,1,-1],[1,2,3,4],[3,4,5,2]]

If A=[(2,-1,1),(-1,2,-1),(1,-1,2)] show that A^(2)-5A+4I=0 Hence find A^(-1)

If A=[{:(1,-1),(2,3):}] , shown that A^(2)-44+5I=o . Hence Find A^(-1) .

If A=[{:(2,-1),(1,3):}] , then show that A^(2)-5A+7I_(2)=O , hence find A^(-1) .