Class XII Maths www.vedantu.com 1`
NCERT Solutions for Class 12
Maths
Chapter 3 - Matrix
Exercise 3.3
1. Find the transpose of each of the following matrices:
(i)
5
1 / 2
-1
Ans: Find transpose
T
51
A 1/ 2 A 5 12
-1
(ii) 1 -1
2 3
Ans: Find transpose
T1 -1 1 2
A A2 3 1 3
(iii)
-1 5 6
3 5 6
2 3 -1
Ans: Find transpose
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T
-1 5 6 -1 3 2
A 3 5 6 A 5 5 3
2 3 -1 6 6 -1
2. If
-1 2 3
A = 5 7 9
-2 1 1
and
-4 1 -5
B = 1 2 0
1 3 1
, then verify that
(i) A + B ' = A' + B'
(ii) A - B ' = A' - B'
Ans: We have
-1 5 -2 -4 1 1
A' 2 7 1 B' 1 2 3
3 9 1 -5 0 1
(i) A + B ' = A' + B'
Ans:
5 3 2 5 6 1
A B 6 9 9 A B ' 3 9 4
1 4 2 2 9 2
-1 5 -2 -4 1 1 -5 6 1
A' B' 2 7 1 1 2 3 3 9 4
3 9 1 -5 0 1 -2 9 2
Hence, we have verified that A B ' A' B'
(ii) A - B ' = A' - B'
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Ans:
3 1 8 3 4 3
A B 4 5 9 A B ' 1 5 2
3 2 0 8 9 0
-1 5 -2 -4 1 1 3 4 3
A' B' 2 7 1 1 2 3 1 5 2
3 9 1 -5 0 1 8 9 0
Hence, we have verified that A - B ' A'- B'
3. If
3 4
A' = -1 2
0 1
and -1 2 1
B =1 2 3
, then verify that
(i) A + B ' = A' + B'
(ii) A - B ' = A' - B'
Ans: We have
1 13 -1 0
A B' 2 24 2 1
1 3
(i) A + B ' = A' + B'
Ans:
2 52 1 1
A B A B ' 1 45 4 4
1 4
3 4 1 1 2 5
A' B' 1 2 2 2 1 4
0 1 1 3 1 4
Hence, we have verified that A B ' A' B'
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(ii) A - B ' = A' - B'
Ans:
4 34 3 1
A B A B ' 3 03 0 2
1 2
3 4 1 1 4 3
A' B' 1 2 2 2 3 0
0 1 1 3 1 2
Hence, we have verified that A - B ' A'- B'
4. If -2 3 -1 0
A' = & B =1 2 1 2
, then find A + 2B '
Ans: Solve for the condition
2 1A
3 2
2 1 1 0 2 1 2 0 4 1A 2B 2
3 2 1 2 3 2 2 4 5 6
4 5(A 2B)
1 6
5. For the matrix A & B , verify that AB ' = B'A'where
1
(i) A = -4 ,B = -1 2 1
3
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Ans:
1 1 2 1
AB 4 1 2 1 4 8 4
3 3 6 3
1 4 3
(AB) 2 8 6
1 4 3
1 1 4 3
B'A' 2 1 4 3 2 8 6
1 1 4 3
Hence, we have verified AB ' B'A'
0
(ii)A = 1 ,B = 1 5 7
2
Ans:
0 0 0 0
AB 1 1 5 7 1 5 7
2 2 10 14
0 1 2
(AB) 0 5 10
0 7 14
1 0 1 2
B'A' 5 0 1 2 0 5 10
7 0 7 14
Hence, we have verified AB ' B'A'
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6. If
cosα sinα(i) A = , then verify thatA'A = I
-sinα cosα
Ans: cos sin
Asin cos
cos sinA'
sin cos
cos sin cos sinA'A
sin cos sin cos
2 2
2 2
cos sin sin cos sin cosA'A
sin cos sin cos sin cos
1 0A'A I
0 1
sinα cosα(ii) A = , then verify thatA'A = I
-cosα sinα
Ans: sin cos
Acos sin
2 2
2 2
sin cosA'
cos sin
sin cos sin cosA'A
cos sin cos sin
cos sin sin cos sin cosA'A
sin cos sin cos sin cos
1 0A'A I
0 1
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7.
(i) Show that the matrix
1 -1 5
A = -1 2 1
5 1 3
is a symmetric matrix
Ans: We have
1 -1 5
A' -1 2 1 A
5 1 3
A' A
Hence, A is a symmetric matrix
(ii) Show that the matrix
0 1 -1
A = -1 0 1
1 -1 0
is a skew symmetric matrix
Ans: We have
0 1 1 0 1 -1
A' 1 0 1 -1 0 1 A
1 1 0 1 -1 0
A' A
Hence, A is a skew-symmetric matrix
8. For the matrix 1 5
A =6 7
, verify that
(i) A + A' is a symmetric matrix
(ii) A + A' is a skew symmetric matrix
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Ans: 1 5 1 6
A ,A'6 7 5 7
(i) A + A' is a symmetric matrix
Ans: 1 5 1 6 2 11
A A'6 7 5 7 11 14
2 11
A A' ' A A'11 14
Hence, A A' is a symmetric matrix
(ii) A + A' is a skew symmetric matrix
Ans: 1 5 1 6 0 1
A A'6 7 5 7 1 0
0 1 0 1
A A' ' A A'1 0 1 0
Hence, A A' is a skew-symmetric matrix
9. Find 1 1
A + A' & A - A'2 2
, when
0 a b
A = -a 0 c
-b -c 0
Ans: The given matrix is
0 a b
A -a 0 c
-b -c 0
, then
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0 a b
A' a 0 c
b c 0
0 a b 0 a b1 1
A A' -a 0 c a 0 c2 2
-b -c 0 b c 0
0 0 01
A A' 0 0 02
0 0 0
0 a b 0 a b1 1
A A' -a 0 c a 0 c2 2
-b -c 0 b c 0
0 a b1
A A' -a 0 c2
-b -c 0
10. Express the following matrices as the sum of a symmetric and a skew
symmetric matrix:
3 5
1 -1
Ans: 3 5 3 11 1
A A'1 -1 5 -12 2
6 61 1A A'
6 22 2
3 31A A'
3 -12
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Thus, 1
A A'2
is a symmetric matrix.
3 5 3 11 1A A'
1 -1 5 -12 2
0 41 1A A'
4 02 2
0 21A A'
2 02
Thus, 1
A A'2
is a skew-symmetric matrix.
11. If A,B are symmetric matrix of same order, then AB - BA is a
A. Skew symmetric matrix
B. Symmetric matrix
C. Zero matrix
D. Identity matrix
Ans: The correct answer is A
A & B are symmetric , therefore , we have
A' A & B' B
Consider
AB BA ' AB ' BA '
B'A ' A 'B'
BA AB
AB BA
AB BA ' AB BA
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Thus, AB BA ' is a skew-symmetric matrix
12. If cosα -sinα
A =sinα cosα
, then A + A' = I , if the value of α is
A. π
6
B. π
3
C. π
D. 3π
2
Ans: The correct answer is B
cos -sinA
sin cos
cos sinA'
sin cos
A A' I
cos -sin cos sin 1 0
sin cos sin cos 0 1
2cos 0 1 0
0 2cos 0 1
3