InterviewSolution
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InterviewSolution
This section includes InterviewSolutions, each offering curated multiple-choice questions to sharpen your knowledge and support exam preparation. Choose a topic below to get started.
| 1. |
Find the inverse of each of the matrices given below : `[(3,-5),(-1,2)]` |
| Answer» Correct Answer - `[(2,5),(1,3)]` | |
| 2. |
Find the inverse of each of the matrices given below : `[(4,1),(2,3)]` |
| Answer» `[{:(" "3/10,-1/10),(-1/5,2/5):}]` | |
| 3. |
Find the inverse of each of the matrices given below : `[(2,-3),(4,6)]` |
| Answer» `[{:((1)/4,1/8),((-1)/6,1/12):}]` | |
| 4. |
Find the inverse of each of the matrices given below : `[(a,b),(c,d)], when (ad-bc)!=0` |
| Answer» `1/(ad-bc).[(d,-b),(-c,a)]` | |
| 5. |
Find the inverse of each of the matrices given below : `[(0,0,-1), (2,-1,4),(-2,-4,-7)]` |
| Answer» `1/4.[(-8,4,4),(11,-2,-3),(-4,0,0)]` | |
| 6. |
Find the inverse of each of the matrices given below : `[(2,-1,4),(-3,0,1),(-1,1,2)]` |
| Answer» `1/19.[(1,-6,1),(-5,-8,14),(3,1,3)]` | |
| 7. |
Find the inverse of each of the matrices given below : `[(8,-4,1),(10,0,6),(8,1,6)]` |
| Answer» `1/10.[(-6,25,-24),(-12,40,-38),(10,-40,40)]` | |
| 8. |
Find the inverse of each of the matrices given below : Compute `(AB)^(-1) when A=[(1,1,2),(0,2,-3),(3,-2,4)] and B^(-1)=[(1,2,0),(0,3,-1),(1,0,2)]`. Find `A^(-1). |
| Answer» `1/19.[(16,12,1),(21,11,-7),(10,-2,3)]` | |
| 9. |
Find the inverse of each of the matrices given below : Obtain the inverse of the matrics `[(1,p,0),(0,1,p),(0,0,1)] and [(1,0,0),(q,1,0),(0,q,1)]`. And, hence find the inverse of the matrix`[((1+pq),p,0),(q,(1+pq),p),(0,q,1)]`. Let the first two matrices be A and B. Then, the third matrix is AB. Now, `(AB)^(-1)=(B^(-1)A^(-1))` |
| Answer» `[(1,-p,p^(2)),(0,1,-p),(0,0,1)],[(1,0,0),(-q,1,0),(q^(2),-q,1)]and[(1,-p,p^(2)),(-q,pq+1,-qp^(2)-p),(q^(2),-pq^(2)-q,p^(2)q^(2)+pq+1)]` | |
| 10. |
Find the inverse of each of the matrices given below : Show that the matrix `A=[(-8,5),(2,4)]` satisfies the equation `x^(2)+4x-42=0` and hence find `A^(-1)`. |
| Answer» `1/42.[(-4,5),(2,8)]` | |
| 11. |
Find the inverse of each of the matrices given below : if `A=[(3,1),(7,5)]` find x and y such that `A^(2)=deltaA-2I`. Hence , find `A^(-1)`. |
| Answer» `x=8,y=8 and A^(-1)=1/8.[(5,7),(-7,3)]` | |
| 12. |
Find the inverse of each of the matrices given below : If `A=[(3,2),(2,1)]`, verify that `A^(2)-4A-I=O, and "hence " "find "A^(-1)`. |
| Answer» Correct Answer - `[(-1,2),(2,-3)]` | |
| 13. |
Find the inverse of each of the matrices given below : If `A=[(3,2),(4,-2)]` , find the value of `delta` so that `A^(2)=deltaA-2I`. Hence, find `A^(-1)`. |
| Answer» `delta=1,A^(-1)=1/2.[(-2,2),(-4,3)]` | |
| 14. |
Show that the matrix, `A=[1 0-2-2-1 2 3 4 1]`satisfies the equation,`A^3-A^2-3A-I_3=O`. Hence, find `A^(-1)`. |
| Answer» `A^(-1)=[(-9,-8,-2),(8,7,2),(-5,-4,-1)]` | |
| 15. |
Find the adjoint of the given matrix and verify in each case that `A.(adj A)=(adj A).A=|A|.I.` `[(cos a,sin a),(sin a,cos a)]` |
| Answer» `[(cos a,-sin a),(-sin a, cos a)]` | |
| 16. |
Find the adjoint of the given matrix and verify in each case that `A.(adj A)=(adj A).A=|A|.I.` `[(1,-1,2),(3,1,-2),(1,0,3)]` |
| Answer» `[(3,3,0),(-11,1,8),(-1,-1,4)]` | |
| 17. |
Find the adjoint of the given matrix and verify in ach case that `A.(adj A)=(adj A).A=|A|.I.` `[(3,-5),(-1,2)]` |
| Answer» Correct Answer - `[(2,5),(1,3)]` | |
| 18. |
Find the adjoint of the given matrix and verify in ach case that `A.(adj A)=(adj A).A=|A|.I.` `[(9,7,3),(5,-1,4),(6,8,2)]` |
| Answer» `[(-34,10,31),(14,0,-21),(46,-30,-44)]` | |
| 19. |
Find the adjoint of the given matrix and verify in ach case that `A.(adj A)=(adj A).A=|A|.I.` `[(4,5,3),(1,0,6),(2,7,9)]` |
| Answer» `[(-42,-24,30),(3,30,-21),(7,-18,-5)]` | |
| 20. |
Find the adjoint of the given matrix and verify in ach case that `A.(adj A)=(adj A).A=|A|.I.` `[(0,1,2),(1,2,3),(3,1,1)]` |
| Answer» `[(-1,1,-1),(8,-6,2),(-5,3,-1)]` | |
| 21. |
Find the adjoint of the given matrix and verify in ach case that `A.(adj A)=(adj A).A=|A|.I.` `[(3,-1,1),(-15,6,-5),(5,-2,2)]` |
| Answer» `[(2,0,-1),(5,1,0),(0,1,3)]` | |
| 22. |
Find the adjoint of the given matrix and verify in each case that `A.(adj A)=(adj A).A=|A|.I.` `[(2,3),(5,9)]` |
| Answer» Correct Answer - `[(9,-3),(-5,2)]` | |
| 23. |
Let `F(a)=[(cos a,-sin a,0),(sin a, cos a, 0),(0,0,1)] and G(B)=[(cos B,0,sin B),(0,1,0),(-sin B,0,cos B)] "Show that " [F(a).G(B)]^(-1)=G(-B).F(-a)`. |
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Answer» We have `F(a).F(-a)=[(cos a, -sin a,0),(sin a, cos a, 0),(0,0,1)] [(cos(-a), -sin (-a),0),(sin (-a), cos (-a),0),(0,0,1)]` `=[(cosa ,-sin a,0),(sin a,cos a, 0),(0,0,1)][(cos a,sin a,0),(-sin a,cos a,0),(0,0,1)]` `[(cos^(2)a+sin^(2)a, 0,0),(0, sin^(2)a+cos^(2)a,0),(0,0,1)] [(1,0,0),(0,1,0),(0,0,1)]=I` `Thus, F(a).F(-a)=I rArr{F(a)}^(-1)=F(-a)`. Similarly, `G(B). G(-B)=I rArr{G(B)}^(-1)=G(-B)`. `:." "{F(a).G(B)}^(-1)={G(B)}^(-1).{F(a)}^(-1) ` [by reversal law] `=G(-B).F(-a)`. Hence,`{F(a).G(B)}^(-1)= G(-B). F(-a)`. |
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| 24. |
If `A=|{:(1,2,2),(2,1,2),(2,2,1):}|`, then show that `A^(2)-4A-5I_(3)=0`. Hemce find `A^(-1)`. |
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Answer» We leave it to the reader to show that `A^(2)-4A-5I=O`. `Now, A^(2)-4A-5I=O` `rArr" "A A-4A=5I` `rArr" "(A A).A^(-1)-4A.A^(-1)=5I.A^(-1)` `rArr" "A(A A^(-1))-4I=5A^(-1)` `rArr" "AI-4I=5A^(-1)` `A-4I=5A^(-1)` `rArr" "A^(-1)=1/5(A-4I)`. `:." "A^(-1)=1/5.{[(1,2,2),(2,1,2),(2,2,1)]-4.[(1,0,0),(0,1,0),(0,0,1)]}` `=1/5.{[(1,2,2),(2,1,2),(2,2,1)]-[(4,0,0),(0,4,0),(0,0,4)]}` `=1/5.[(-3,2,2),(2,-3,2),(2,2,-3)]`. |
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| 25. |
Find the inverse of the matix, `A=[[2,-3],[-4,7]]`. |
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Answer» We have `|A|=|(2,-3),(-4,7)|=(14-12)=2!=0` So, `A_(1)` exists. The cofactors of the elements of `|A|` are given by `A_(11)=7,A_(12)=-(-4)=4,` `A_(21)=-(-3)=3, A_(22)=2.` `:." "(adj A)=[[7,4],[3,2]]=[[7,3],[4,2]].` `Hence,A^(-1)=1/|A|.(adj A)` `=1/2.[[7,3],[4,2]]=[[7/2,3/2],[2,1]]` |
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