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Choose the correct answer in the Exercises 11 and 12.

Q12.    If A = \begin{bmatrix} \cos\alpha & -\sin\alpha\\ \sin\alpha& \cos\alpha \end{bmatrix} and A+A' =I, then the value of \alpha is

            (A)    \frac{\pi}{6}

            (B)    \frac{\pi}{3}

            (C)    \pi

            (D)    \frac{3\pi}{2}

  Option B is correct.    

Choose the correct answer in the Exercises 11 and 12.

Q11.    If A, B are symmetric matrices of same order, then AB – BA is a

            (A) Skew symmetric matrix
            (B) Symmetric matrix
            (C) Zero matrix
            (D) Identity matrix

  If A, B are symmetric matrices then                                                          and       we have,                                                                                                                                                                                                                                 Hence, we have  Thus,( AB-BA)' is skew symmetric. Option A...

Q10.    Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

            (iv)    \begin{bmatrix} 1 & 5\\ -1 & 2 \end{bmatrix}

Let                       Thus,    is a symmetric matrix.                Let  Thus,  is a skew-symmetric matrix. Represent   A  as the sum of B and C.      

Q10.    Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

            (iii)    \begin{bmatrix} 3 & 3 & -1\\ -2 & -2 & 1\\ -4 & -5 & 2 \end{bmatrix}

Let                       Thus,    is a symmetric matrix.                Let  Thus,  is a skew-symmetric matrix. Represent   A  as the sum of B and C.    

Q10.    Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

            (ii)    \begin{bmatrix} 6 & -2 & 2\\ -2 & 3 & -1\\ 2 & -1 & 3 \end{bmatrix}

Let                       Thus,    is a symmetric matrix.                Let  Thus,  is a skew-symmetric matrix. Represent   A  as the sum of B and C.    

10.    Express the following matrices as the sum of a symmetric and a skew symmetric matrix:

            (i)    \begin{bmatrix} 3 & 5\\ 1 & -1 \end{bmatrix}

Let                       Thus,    is a symmetric matrix.                Let  Thus,  is a skew symmetric  matrix. Represent   A  as sum of B and C.                  

Q9.    Find \frac{1}{2}(A+A') and \frac{1}{2}(A-A'), when A = \begin{bmatrix} 0 & a & b\\ -a & 0 & c\\ -b & -c & 0 \end{bmatrix}

the transpose of the matrix is obtained by interchanging rows and columns             

Q8.    For the matrix A = \begin{bmatrix} 1 & 5\\ 6 & 7 \end{bmatrix}, verify that

            (ii) (A - A') is a skew symmetric matrix.

   We have  Hence ,   is a skew-symmetric matrix.

Q8.    For the matrix A = \begin{bmatrix} 1 & 5\\ 6 & 7 \end{bmatrix}, verify that

            (i) (A + A') is a symmetric matrix.

   We have  Hence ,   is a symmetric matrix.

Q7.    (ii) Show that the matrix A = \begin{bmatrix} 0 & 1 & -1\\ -1 & 0 &1 \\ 1 & -1 &0 \end{bmatrix} is a skew-symmetric matrix.

The transpose of A is Since, so given matrix is a skew-symmetric matrix.

Q7.    (i) Show that the matrix A = \begin{bmatrix} 1 &- 1& 5\\ -1 & 2 & 1\\ 5 & 1 & 3 \end{bmatrix} is a symmetric matrix.

the transpose of A is Since, so given matrix is a symmetric matrix.        

Q6.    (ii) If A = \begin{bmatrix} \sin\alpha & \cos\alpha \\ -\cos\alpha & \sin\alpha \end{bmatrix}, then verify that A'A = I

By interchanging columns and rows of the matrix A we get the transpose of A To prove:  L.H.S :  

Q6.    (i) If A = \begin{bmatrix} \cos\alpha & \sin\alpha \\ -\sin\alpha & \cos\alpha \end{bmatrix}, then verify that A'A =I

By interchanging rows and columns we get transpose of A To prove:  L.H.S :        

Q5.    For the matrices A and B, verify that (AB)' = B'A', where

            (ii)    A = \begin{bmatrix} 0\\ 1\\ 2 \end{bmatrix}B = \begin{bmatrix} 1 & 5&7 \end{bmatrix}

   ,  To prove :  Heence, L.H.S =R.H.S  i.e..

Q5.    For the matrices A and B, verify that (AB)' = B'A', where

            (i)    A = \begin{bmatrix} 1\\-4 \\3 \end{bmatrix}B = \begin{bmatrix} -1& 2 &1 \end{bmatrix}

  ,     To prove :  Hence, L.H.S =R.H.S  so it is verified that .          

Q4.    If A' = \begin{bmatrix} -2 & 3\\ 1 & 2 \end{bmatrix} and B= \begin{bmatrix} -1 & 0\\ 1 & 2 \end{bmatrix}, then find (A + 2B)'

 : Transpose is obtained by interchanging rows and columns and the transpose of A+2B is

Q3.    If A = \begin{bmatrix} 3 & 4\\ -1 &2 \\ 0 & 1 \end{bmatrix} and B = \begin{bmatrix} -1 & 2 & 1\\ 1 &2 &3 \end{bmatrix}, then verify

            (ii)    (A - B)' = A' - B'

      To prove:        R.H.S:      Hence, L.H.S = R.H.S i.e. .

Q3.    If A' = \begin{bmatrix} 3 & 4\\ -1 &2 \\ 0 & 1 \end{bmatrix}  and B = \begin{bmatrix} -1 & 2 & 1\\ 1 &2 &3 \end{bmatrix}, then verify

            (i)    (A + B)' = A' + B'

A' = \begin{bmatrix} 3 & 4\\ -1 &2 \\ 0 & 1 \end{bmatrix}     B = \begin{bmatrix} -1 & 2 & 1\\ 1 &2 &3 \end{bmatrix}

A=(A')' = \begin{bmatrix} 3 & -1&0\\ 4 &2 & 1 \end{bmatrix}

To prove: (A + B)' = A' + B'

L.H.S : (A + B)' = 

A+B = \begin{bmatrix} 3 & -1&0\\ 4 &2 & 1 \end{bmatrix}  + \begin{bmatrix} -1 & 2 & 1\\ 1 &2 &3 \end{bmatrix} 

A+B = \begin{bmatrix} 3+(-1) & -1+(-1)&0+1\\ 4+1 &2+2 & 1+3 \end{bmatrix}

A+B = \begin{bmatrix} 2 & -2&1\\ 5 &4 & 4 \end{bmatrix}

\therefore \, \, \, (A+B)' = \begin{bmatrix} 2 & 5\\ 1 &4\\1 & 4 \end{bmatrix}

R.H.S:  A' + B'

A'+B' = \begin{bmatrix} 3 & 4\\ -1 &2 \\ 0 & 1 \end{bmatrix}  + \begin{bmatrix} -1 & 1\\ 2 &2 \\ 1 & 3 \end{bmatrix}

A'+B' = \begin{bmatrix} 2 & 5\\ 1 &4 \\ 1 & 4 \end{bmatrix}

Hence, L.H.S = R.H.S, hence verified that (A + B)' = A' + B'.

 

 

 

 

 

 

 

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Q2.    If A = \begin{bmatrix} -1 & 2 & 3\\ 5 &7 &9 \\ -2 & 1 & 1 \end{bmatrix} and B = \begin{bmatrix} -4 & 1 & -5\\ 1 &2 &0 \\ 1 & 3 & 1 \end{bmatrix}, then verify

            (ii)    (A - B)' = A' - B'

      and        L.H.S :        R.H.S :      Hence, L.H.S = R.H.S. so verified that   .

Q2.    If A = \begin{bmatrix} -1 & 2 & 3\\ 5 &7 &9 \\ -2 & 1 & 1 \end{bmatrix} and B = \begin{bmatrix} -4 & 1 & -5\\ 1 &2 &0 \\ 1 & 3 & 1 \end{bmatrix}, then verify

            (i)    (A + B)' = A' + B'

      and        L.H.S :        R.H.S :      Thus we find that the LHS is equal to RHS and hence verified.                         
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