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.

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