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If A is a square matrix then,

Option: 1

A+A' is symmetric matrix


Option: 2

A-A' is skew-symmetric matrix


Option: 3

A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')


Option: 4

All of the above


Answers (1)

best_answer

 

 

Properties of symmetric and skew symmetric matrices -

Properties of symmetric and skew-symmetric matrices:

     i) if A is a square matrix, then AA’ and A’A are symmetric matrices

    ii) All positive integral power of symmetric matrices are symmetric matrices, because 

      (A^n)'=(A')^n

    iii) If A is a symmetric matrix, then -A, kA, A’, An, A-1, B’AB are also symmetric matrix where n ∈ N, k ∈ R and is B a         square matrix of order same as matrix A.

    iv) If A is a skew-symmetric matrix then   

  1. A2n is a symmetric matrix for n ? N.

  2. A2n+1 is a skew-symmetric matrix for n ? N

  3. kA is also a skew-symmetric matrix, where  k ∈ R

  4. B’AB  is also a skew-symmetric matrix where B a square matrix of order same as matrix A

    v) If A and B are symmetric matrices then:

  1. A ± B, AB+BA are symmetric matrices.

  2. AB - BA is a skew-symmetric matrix.

  3. AB is a symmetric matrix iff AB = BA.

    iv) If A and B are skew-symmetric matrices then:

  1. A ± B, AB - BA are skew-symmetric matrices.

  2. AB + BA is a symmetric matrix.

    vi) If A is a skew-symmetric matrix and C is a column matrix, then C’AC is a zero matrix.

-

 

 

Let  A=\begin{bmatrix} a & b\\ c & d \end{bmatrix} 

now, A'=\begin{bmatrix} a & c\\ b & d \end{bmatrix}

A+A'=\begin{bmatrix} 2a & b+c\\ b+c & 2d \end{bmatrix} which is a symmetric matrix

A-A'=\begin{bmatrix} 0 & b-c\\ -(b-c) & 0 \end{bmatrix}, which is a skew-symmetric matrix

A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')\\\\ A=\frac{1}{2}\begin{bmatrix} 2a & b+c\\ b+c & 2d \end{bmatrix}+\frac{1}{2}\begin{bmatrix} 0 & b-c\\ -(b-c) & 0 \end{bmatrix}\\\\A=A'=\begin{bmatrix} a & c\\ b & d \end{bmatrix}

Hence all the options are correct

 

Posted by

Sanket Gandhi

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