Practicing ISC S Chand Maths Class 12 Solutions Chapter 6 Matrices Ex 6(f) is the ultimate need for students who intend to score good marks in examinations.
S Chand Class 12 ICSE Maths Solutions Chapter 6 Matrices Ex 6(f)
Question 1.
Define a symmetric matrix. Show that the following matrices are symmetric.
(i) \(\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]\)
(ii) \(\left[\begin{array}{ccc}
2+i & 1 & 3 \\
1 & 2 & 3+2 i \\
3 & 3+2 i & 4
\end{array}\right]\)
Solution:
A matrix P is said to symmetric iff P’ = P
(i) Let P = \(\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]\)
∴ P’ = \(\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]^{\prime}=\left[\begin{array}{lll}
a & h & g \\
h & b & f \\
g & f & c
\end{array}\right]\) = P
Thus P is symmetric.
(ii) Let P = \(\left[\begin{array}{ccc}
2+i & 1 & 3 \\
1 & 2 & 3+2 i \\
3 & 3+2 i & 4
\end{array}\right]\)
∴ P’ = \(\left[\begin{array}{ccc}
2+i & 1 & 3 \\
1 & 2 & 3+2 i \\
3 & 3+2 i & 4
\end{array}\right]\)
= \(\left[\begin{array}{ccc}
2+i & 1 & 3 \\
1 & 2 & 3+2 i \\
3 & 3+2 i & 4
\end{array}\right]\) = P
Thus P is symmetric.
Question 2.
Show that each of the following is a skew matrix.
(i) \(\left[\begin{array}{rrr}
0 & 5 & -4 \\
-5 & 0 & 7 \\
4 & -7 & 0
\end{array}\right]\)
(ii) \(\left[\begin{array}{rrr}
a & b & -c \\
-b & 0 & d \\
c & -d & o
\end{array}\right]\)
(iii) \(\left[\begin{array}{rrr}
0 & 2 i & 3 \\
-2 i & 0 & 4 \\
-3 & -4 & 0
\end{array}\right]\)
Solution:
(i) Let Q = \(\left[\begin{array}{rrr}
0 & 5 & -4 \\
-5 & 0 & 7 \\
4 & -7 & 0
\end{array}\right]\)
∴ Q’ = \(\left[\begin{array}{rrr}
0 & -5 & 4 \\
5 & 0 & -7 \\
-4 & 7 & 0
\end{array}\right]\)
= – \(\left[\begin{array}{rrr}
0 & 5 & -4 \\
-5 & 0 & 7 \\
4 & -7 & 0
\end{array}\right]\) = – Q
Thus, Q is skew symmetric.
(ii) Let Q = \(\left[\begin{array}{rrr}
0 & b & -c \\
-b & 0 & d \\
c & -d & 0
\end{array}\right]\)
∴ Q’ = \(\left[\begin{array}{rrr}
0 & -b & c \\
b & 0 & -d \\
-c & d & 0
\end{array}\right]\)
= – \(\left[\begin{array}{rrr}
0 & b & -c \\
-b & 0 & d \\
c & -d & 0
\end{array}\right]\) = – Q
Thus, Q is skew symmetric.
(iii) Let Q = \(\left[\begin{array}{rrr}
0 & 2 i & 3 \\
-2 i & 0 & 4 \\
-3 & -4 & 0
\end{array}\right]\)
∴ Q’ = \(\left[\begin{array}{rrr}
0 & -2 i & -3 \\
2 i & 0 & -4 \\
3 & 4 & 0
\end{array}\right]\)
= – \(\left[\begin{array}{rrr}
0 & 2 i & 3 \\
-2 i & 0 & +4 \\
-3 & -4 & 0
\end{array}\right]\) = – Q
Thus, Q is skew symmetric.
Question 3.
Show that A + A’ is a symmetric matrix if
(i) A = \(\left[\begin{array}{ll}
4 & 1 \\
5 & 8
\end{array}\right]\) (ii) \(\left[\begin{array}{ll}
2 & 4 \\
5 & 6
\end{array}\right]\)
Solution:
Question 4.
Show that A – A’ is a skew-symmetric matrix if
(i) A = \(\left[\begin{array}{ll}
3 & 4 \\
5 & 1
\end{array}\right]\)
(ii) A = \(\left[\begin{array}{ll}
3 & -4 \\
1 & -1
\end{array}\right]\)
(iii) A = \(\left[\begin{array}{ll}
1 & 4 \\
3 & 7
\end{array}\right]\)
Solution:
(i) Let A = \(\left[\begin{array}{ll}
3 & 4 \\
5 & 1
\end{array}\right]\)
∴ A’ = \(\left[\begin{array}{ll}
3 & 5 \\
4 & 1
\end{array}\right]\)
Let P = A – A’ = \(\left[\begin{array}{ll}
3 & 4 \\
5 & 1
\end{array}\right]-\left[\begin{array}{ll}
3 & 5 \\
4 & 1
\end{array}\right]\)
= \(\left[\begin{array}{cc}
3-3 & 4-5 \\
5-4 & 1-1
\end{array}\right]=\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]\)
∴ P’ = \(\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]^{\prime}=\left[\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right]\)
= – \(\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]\) = – P
Thus P is skew symmetric matrix.
(ii) Let A = \(\left[\begin{array}{ll}
3 & -4 \\
1 & -1
\end{array}\right]\)
∴ A’ = \(\left[\begin{array}{rr}
3 & 1 \\
-4 & -1
\end{array}\right]\)
Let Q = A – A’ = \(\left[\begin{array}{rr}
3 & -4 \\
1 & -1
\end{array}\right]-\left[\begin{array}{rr}
3 & 1 \\
-4 & -1
\end{array}\right]\)
= \(\left[\begin{array}{cc}
3-3 & -4-1 \\
1+4 & -1+1
\end{array}\right]=\left[\begin{array}{cc}
0 & -5 \\
5 & 0
\end{array}\right]\)
∴ Q’ = \(\left[\begin{array}{cc}
0 & 5 \\
-5 & 0
\end{array}\right]=-\left[\begin{array}{cc}
0 & -5 \\
5 & 0
\end{array}\right]\) = – Q
Thus Q is skew symmetric matrix.
(iii) Let A = \(\left[\begin{array}{ll}
1 & 4 \\
3 & 7
\end{array}\right]\)
∴ A’ = \(\left[\begin{array}{ll}
1 & 3 \\
4 & 7
\end{array}\right]\)
Let Q = A – A’ = \(\left[\begin{array}{ll}
1 & 4 \\
3 & 7
\end{array}\right]-\left[\begin{array}{ll}
1 & 3 \\
4 & 7
\end{array}\right]\)
= \(\left[\begin{array}{cc}
1-1 & 4-3 \\
3-4 & 7-7
\end{array}\right]=\left[\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right]\)
∴ P’ = \(\left[\begin{array}{cc}
0 & -1 \\
1 & 0
\end{array}\right]=-\left[\begin{array}{cc}
0 & 1 \\
-1 & 0
\end{array}\right]\) = – P
Thus P i.e. A – A’ is skew symmetric matrix.
Question 5.
If A = \(\left[\begin{array}{cc}
4 & x+2 \\
2 x-3 & x+1
\end{array}\right]\) is a symmetric matrix, find x.
Solution:
Given A = \(\left[\begin{array}{cc}
4 & x+2 \\
2 x-3 & x+1
\end{array}\right]\)
∴ A’ = \(\left[\begin{array}{cr}
4 & 2 x-3 \\
x+2 & x+1
\end{array}\right]\)
Since A be symmetric matrix. ∴ A’ = A
\(\left[\begin{array}{cr}
4 & 2 x-3 \\
x+2 & x+1
\end{array}\right]=\left[\begin{array}{cc}
4 & x+2 \\
2 x-3 & x+1
\end{array}\right]\)
Thus their corresponding elements are equal.
∴ 2x – 3 = x + 2 ⇒ x = 5
x + 2 = 2x – 3 ⇒ x = 5
Thus, x = 5
Question 6.
Express the following as the sum of a symmetric and skew symmetric matrix.
(i) \(\left[\begin{array}{ll}
7 & 4 \\
5 & 3
\end{array}\right]\)
(ii) \(\left[\begin{array}{ll}
3 & 2 \\
4 & 5
\end{array}\right]\)
(iii) \(\left[\begin{array}{rrr}
2 & 4 & -6 \\
7 & 3 & 5 \\
1 & -2 & 4
\end{array}\right]\)
Solution:
Question 7.
Find the symmetric and skew symmetric parts of the matrix
A = \(\left[\begin{array}{lll}
1 & 2 & 4 \\
6 & 8 & 1 \\
3 & 5 & 7
\end{array}\right]\)
Solution:
Question 8.
Let A = \(\left[\begin{array}{rrr}
3 & 2 & 7 \\
1 & 4 & 3 \\
-2 & 5 & 8
\end{array}\right]\), find X and Y such that X + Y = A and X is a symmetric and Y a skew symmetric matrix.
Solution:
Question 9.
Express the following matrix as the sum of a symmetric and a skew symmetric matrix, and verify your result \(\left[\begin{array}{rrr}
3 & -2 & -4 \\
3 & -2 & -5 \\
-1 & 1 & 2
\end{array}\right]\)
Solution:
Let A = \(\left[\begin{array}{rrr}
3 & -2 & -4 \\
3 & -2 & -5 \\
-1 & 1 & 2
\end{array}\right]\)
Thus, Q is skew symmetric matrix.
∴ from (1); we canclude that, matrix A be the sum of a symmetric and a skew symmetric matrix.