If `A` and `B` are matrices of the same order, then `A B^T-B^T A` is a (a) skew-symmetric matrix (b) null matrix (c) unit matrix (d) symmetric matrix
1 Answers
Talk Doctor Online in Bissoy App
If a matrix A is both symmetric and skew symmetric, then A is necessarily a :
Option : (b) Zero square matrix
2 Answers 1 views
If A is symmetric as well as skew-symmetric matrix, then A is
Correct Answer - B Let `A=[a_("ij")]`. Since a is skew-symmetric, we have `a_(ii)=0` and `a_("ij")=-a_("ij") (i ne j)` A is symmetric as well, so `a_("ij")=a_("ji")` for all `i` and `j`. `:....
2 Answers 1 views
Let A and B be two square matrices of the same size such that `AB^(T)+BA^(T)=O`. If A is a skew-symmetric matrix then BA is
Correct Answer - B Let `C=BA`, then `C^(T)=(BA)^(T)=A^(T)B^(T)` `=-AB^(T)` (as A is skew-symmetric) `=BA^(T)" "("as "AB^(T)+BA^(T)=O)` `=B (-A)` `=- BA=-C` Therefore, C is skew-symmetric.
2 Answers 1 views
Let `aa n db` be two real numbers such that `a >1,b > 1.` If `A=(a0 0b)` , then `(lim)_(nvecoo)A^(-n)` is a. unit matrix b. null matrix c. `2l` d. non
Correct Answer - B `A=((a,0),(0,b))` `implies A^(2)=((a,0),(0,b))((a,0),(0,b))=((a^(2),0),(0,b^(2)))` `implies A^(3)=((a^(2),0),(0,b^(2))) ((a,0),(0,b))=((a^(3),0),(0,b^(3)))` `implies A^(n)=((a^(n),0),(0,b^(n)))` `implies (A^(n))^(-1) =1/(a^(n)b^(n)) ((b^(n),0),(0,a^(n)))=((a^(-n),0),(0,b^(-n)))` `implies lim_(n rarr oo) (A^(n))^(-1)=((0,0),(0,0))` as `a gt 1` and `b gt 1`
2 Answers 1 views
If Ais symmetric and B is skew-symmetric matrix, then which of the following is/are CORRECT ?
`A^(T)=A` and `B^(T)=-B` (1) `(ABA^(T))^(T)=AB^(T)A^(T)=-(ABA^(T))` (2) `AB^(T)+BA^(T)=-AB+BA` Now, `(BA-AB)^(T)=(BA)^(T)-(AB)^(T)` `=A^(T)B^(T)-B^(T)A^(T)=-AB+BA=AB^(T)+BA^(T)` (3) `((A+B)(A-B))^(T)=(A-B)^(T) (A+B)^(T)` `=(A^(T)-B^(T)) (A^(T)+B^(T))=(A+B) (A-B)` (4) `((A+I) (B-I))^(T)=(B^(T)-I) (A^(T)+I)` `=-(-B+I) (A+I)=(B-I) (A+I)`
2 Answers 2 views
Let A and B two symmetric matrices of order 3. Statement 1 : `A(BA)` and `(AB)A` are symmetric matrices. Statement 2 : `AB` is symmetric matrix if mat
Correct Answer - C `(A(BA))^(T)=(BA)^(T) A^(T)` `=(A^(T) B^(T)).A^(T)` `=(AB) A` `=A(BA)` Similarly `((AB)A)^(T)=(AB)A` Clearly both statements are true but statement 2 is not a correct explanation of statement 1.
2 Answers 1 views
`A,B,C` are three matrices of the same order such that any two are symmetric and the `3^(rd)` one is skew symmetric. If `X=ABC+CBA` and `Y=ABC-CBA`, t
Correct Answer - D `(d)` `(XY)^(T)=Y^(T)X^(T)` `Y^(T)=(ABC-CBA)^(T)` `=C^(T)B^(T)A^(T)-A^(T)B^(T)C^(T)` `=-CBA+ABC=Y` `X^(T)=(ABC+CBA)^(T)` `=C^(T)B^(T)A^(T)+A^(T)B^(T)C^(T)` `=-CBA-ABC=-X`
2 Answers 1 views
If `A`, `B` are two square matrices of same order such that `A+B=AB` and `I` is identity matrix of order same as that of `A`,`B` , then
Correct Answer - A::C `(a,c)` `A+B=AB` `impliesI-(A+B-AB)=I` `implies(I-A)(I-B)=I` `implies|I-A||I-B|=I` `implies |I-A|`, `|I-B|` are non zero Also `(I-B)(I-A)=I` `impliesI-B-A+BA=I` `impliesA+B=B+A` `impliesAB=BA` `implies(a)` and `(c )` are correct
2 Answers 1 views
If `A` is a skew symmetric matrix, then `B=(I-A)(I+A)^(-1)` is (where `I` is an identity matrix of same order as of `A`)
Correct Answer - C `(c )` `B=(I-A)(I+A)^(-1)` `impliesB^(T)=(I+A^(T))^(-1)(I-A^(T))` `=(I-A)^(-1)(I+A)` `impliesB B^(T)=(I-A)(I+A)^(-1)(I-A)^(-1)(I+A)` `=(I-A)(I-A)^(-1)(I+A)^(-1)(I+A)` `=I` (As `(I-A).(I+A)=(I+A)(I-A)`)
2 Answers 2 views
Matrices of order `3xx3` are formed by using the elements of the set `A={-3,-2,-1,0,1,2,3}`, then probability that matrix is either symmetric or skew
Correct Answer - D `(d)` For symmetric matrix each place in upper triangle and leading diagonal can be filled in `7` ways. Then number of symmetric matrices are `7^(6)`. For skew...
2 Answers 2 views