 Mathematics 9 (Complete Solutions)

# Q8 Verify that

If
A =

 1 2 0 1

,
B =

 1 1 2 0

, then verify that

 (i) (A + B)t = At + Bt (ii) (A - B)t = At - Bt (iii) A + At is symmetric (iv) A - At is skew symmetric (v) B + Bt is symmetric (vi) B - Bt is skew symmetric

Solution:

(i) (A + B)t = At + Bt
(

 1 2 0 1

+

 1 1 2 0

)t
=
(

 1 2 0 1

)t
+
(

 1 1 2 0

)t
(

 1+1 2+1 0+2 1+0

)t
=

 1 0 2 1

+

 1 2 1 0

(

 2 3 2 1

)t
=

 1+1 0+2 2+1 1+0

 2 2 3 1

=

 2 2 3 1

Hence proved L.H.S = R.H.S

(ii) (A - B)t = At - Bt
(

 1 2 0 1

-

 1 1 2 0

)t
=
(

 1 2 0 1

)t
-
(

 1 1 2 0

)t
(

 1-1 2-1 0-2 1-0

)t
=

 1 0 2 1

-

 1 2 1 0

(

 0 1 -2 1

)t
=

 1-1 0-2 2-1 1-0

 0 -2 1 1

=

 0 -2 1 1

Hence proved L.H.S = R.H.S

(iii) A + At is symmetric
A + At =

 1 2 0 1

+
(

 1 2 0 1

)t
A + At =

 1 2 0 1

+

 1 0 2 1

A + At =

 1+1 2+0 0+2 1+1

A + At =

 2 2 2 2

(A + At)t =

 2 2 2 2

So A + At is a symmetric matrix.

(iv) A - At is skew symmetric
A - At =

 1 2 0 1

-
(

 1 2 0 1

)t
A - At =

 1 2 0 1

-

 1 0 2 1

A - At =

 1-1 2-0 0-2 1-1

A - At =

 0 2 -2 0

(A - At)t =

 0 -2 2 0

So A - At is a skew symmetric matrix.

(v) B + Bt is symmetric
B + Bt =

 1 1 2 0

+
(

 1 1 2 0

)t
B + Bt =

 1 1 2 0

+

 1 2 1 0

B + Bt =

 1+1 1+2 2+1 0+0

B + Bt =

 2 3 3 0

(B + Bt)t =

 2 3 3 0

So B + Bt is a symmetric matrix.

(vi) B - Bt is skew symmetric
B - Bt =

 1 1 2 0

-
(

 1 1 2 0

)t
B - Bt =

 1 1 2 0

-

 1 2 1 0

B - Bt =

 1-1 1-2 2-1 0-0

B - Bt =

 0 -1 1 0

(B - Bt)t =

 0 1 -1 0

So B - Bt is a skew symmetric matrix.