Let B is an invertible square matrix and B is the adjoint of matrix A such that `AB=B^(T)`. Then

Monjur Alahi

Let B is an invertible square matrix and B is the adjoint of matrix A such that `AB=B^(T)`. Then
A. A is an identity matrix
B. B is symmetric matrix
C. A is a skew-symmetric matrix
D. B is skew symmetic matrix


0 like 0 dislike
2 views
Share with your friends

1 Answers

Salim Ahmed Kawsar

Call

Given that `AB=B^(T)` and `B=` adj. A
`implies A=B^(T)B^(-1)`
`implies |A|=|B^(T)|xx|B^(-1)|=1` ...(i)
Now, adj. `A=` adj `(B^(T) B^(-1))`
`="adj"(B^(-1))xx"adj"(B^(T))`
`=("adj B")^(-1)xx"adj"(B^(T))`
`implies ("adj. B")^(T)=("adj. B")xx("adj. A")=("adj. B")B=|B|I`
`implies ("adj. B")^(T)=|B|. I`
`implies ("adj. (adj. A)")^(T)=|"adj. A"|I`
`:. |A|^(n-2) A^(T)=|A|^(n-1)I`
`implies A^(T)=|A|I`
`implies A^(T)=I`
`implies A=I`
`:.` B=adj. A=adj. `I=I`

0 like 0 dislike
2 views
Talk Doctor Online in Bissoy App
Matrix A ha s m rows and n+ 5 columns; matrix B has m rows and `11 - n` columns. If both AB and BA exist, then (A) AB and BA are square matrix (B) AB

If both AB and BA exist, then the number of columns of A is equal to the number of rows of B. Therefore, `n+5=m` (1) And the number of columns...

2 Answers 1 views
Let `A` be a square matrix of order 3 such that adj. (adj. (adj. A)) `=[(16,0,-24),(0,4,0),(0,12,4)]`. Then find (i) `|A|` (ii) adj. A

We know that adj. (adj. A) `=|A|^(n-2)A`, where n is order of matrix. `:.` adj. (adj. (adj. A))`=|"adj. A"|^(n-2)` adj. A `=(|A|^(n-1))^((n-2))` adj. A For `n=3`, adj. (adj. (adj. A))`=|A|^(2)`...

2 Answers 1 views
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

Let `P=(AB^(T)-BA^(T))` `:. P^(T)=(AB^(T)-BA^(T))^(T)=(AB^(T))^(T)-(BA^(T))^(T)` `=(B^(T))^(T) (A)^(T)-(A^(T))^(T)B^(T)=BA^(T)-AB^(T)` `=-(AB^(T)-BA^(T))=-P` Hence, `(AB^(T)-BA^(T))` is a skew-symmetric matric.

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
Let A be an `mxxn` matrix. If there exists a matrix L of type `nxxm` such that `LA=I_(n)`, then L is called left inverse of A. Similarly, if there exi

Correct Answer - C As second row of all the options is same, we have to look at the elements of the first row. Let the left inverse be `[(a,b,c),(d,e,f)]`. Then...

2 Answers 1 views
Let P, Q and R be invertible matrices of order 3 such `A=PQ^(-1), B=QR^(-1)` and `C=RP^(-1)`. Then the value of det. `(ABC+BCA+CAB)` is equal to _____

Correct Answer - 27 We have `A=PQ^(-1), B=QR^(-1), C=RP^(-1)` `:. ABC=I, BCA=I, CAB=I` `:.` det. `(ABC+BCA+CAB)=` det. `(3I)=3^(3)xx` det. `(I)=27`

2 Answers 1 views
Let `M` be a `2xx2` symmetric matrix with integer entries. Then `M` is invertible if The first column of `M` is the transpose of the second row of `M`

Correct Answer - C::D Let `M=[(a,c),(c,b)]" "("where "a, b, c, in I)` (1) If the first column of M is the transpose of the second row of M, then `[(a,c)]=[(c,b)]` `:....

2 Answers 1 views
Let `A` be a square matrix of order `3` so that sum of elements of each row is `1`. Then the sum elements of matrix `A^(2)` is

Correct Answer - B `(b)` Let `A=[{:(a,b,c),(p,q,r),(x,y,z):}]` Given `{:(a+b+c=1),(p+q+r=1),(x+y+z=1):}` `impliesA^(2)=[{:(a,b,c),(p,q,r),(x,y,z):}][{:(a,b,c),(p,q,r),(x,y,z):}]` `=[{:(a^(2)+bp+cx,,ab+bq+cy,,ac+br+cz),(pa+qp+rx,,qb+q^(2)+ry,,pc+qr+rz),(xa+yp+zx,,xb+yq+zy,,xc+yr+z^(2)):}]` Sum of elements of `R_(1)=a^(2)+bp+cx+ab+bq+cy+ac+br+cz` `=a(a+b+c)+b(p+q+r)+c(x+y+z)` `=a+b+c=0` Similarly sum of elements of `R_(2)=p(a+b+c)+q(p+q+r)+r(x+y+z)` `=p+q+r=1` `R_(3)=x(a+b+c)+y(p+q+r)+z(x+y+z)` `=x+y+z=1` `:.` sum of element...

2 Answers 1 views
A be a square matrix of order `2` with `|A| ne 0` such that `|A+|A|adj(A)|=0`, where `adj(A)` is a adjoint of matrix `A`, then the value of `|A-|A|adj

Correct Answer - D `(d)` Let `A=[{:(m,n),(p,q):}]`,`adj(a)=[{:(q,-n),(-p,m):}]` Let `|A|=d=mq-np` `|A+dadjA|=|{:(m+qd,n(1-d)),(p(1-d),q+md):}|=0` `impliesmq+m^(2)d+q^(2)d+mqd^(2)-np+2npd-npd^(2)=0` `implies(mq-np)+(mq0np)d^(2)+m^(2)d+q^(2)d+2mqd-2d^(2)=0` `implies(d+d^(3)-2d^(2))+d(m^(2)+q^(2)+2mq)=0` Now, `|A- d adj|A|=-(m+q)^(2)+4(mq-np)=4d=4`

2 Answers 1 views
X