Home
Class 12
MATHS
If the elements of a matrix A are real p...

If the elements of a matrix `A` are real positive and distinct such that `det(A+A^(T))^(T)=0` then

A

`detA gt 0`

B

`det A ge 0`

C

`det (A-A^(T)) gt 0`

D

`det (A.A^(T)) gt 0`

Text Solution

Verified by Experts

The correct Answer is:
A, C, D
`(a,c,d)` `A=({:(a,b),(c,d):})` `a ne b ne c ne d gt 0`
`A+A^(T)=({:(2a,b+c),(b+c,2d):})`
`|A+A^(T)|=4ad-(b+c)^(2)=0impliesb+c=2sqrt(ad)`
`impliesb+c=2sqrt(ad) gt2sqrt(bc)` `(A.M gt G.M)`
`impliesad gt bc`
`impliesad-bc gt 0` (as `a ne b ne c ne d gt 0`)
`impliesdetA gt0`
`|A-A^(T)|=|{:((0,b-c),(c-b,0)):}|=0+(b-c)^(2) gt 0`
`|A A^(T)|=|A||A^(T)|=|A|^(2)=(det A)^(2) gt 0`
Promotional Banner

Topper's Solved these Questions

  • MATRICES

    CENGAGE|Exercise Single correct Answer|34 Videos

  • MATHMETICAL REASONING

    CENGAGE|Exercise JEE Previous Year|10 Videos

  • METHODS OF DIFFERENTIATION

    CENGAGE|Exercise Multiple Correct Answer Type|7 Videos

Similar Questions

Explore conceptually related problems

Let A be any 3xx2 matrix. Then prove that det. (A A^(T))=0 .

If A^(T) is the transpose of a square matrix A, then

If one of the eigenvalues of a square matrix a order 3xx3 is zero, then prove that det A=0 .

If A is a 3xx4 matrix and B is a matrix such that both A^(T) B and BA^(T) are defined, what is the order of the matrix B?

let f(x)=2+cosx for all real x Statement 1: For each real t, there exists a pointc in [t,t+pi] such that f'(c)=0 Because statement 2: f(t)=f(t+2pi) for each real t

Let A be an orthogonal matrix, and B is a matrix such that AB=BA , then show that AB^(T)=B^(T)A .

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

If A and B are square matrix of same order then (A-B)^(T) is:

If A is a matrix of order 3, then det (kA) …………….