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Example 1. If A is a skew-symmetric matrix and X is a column matrix
then show that X’ AX is a null matrix.
Solution: Since A is a skew-symmetric matrix A’=
(M.U.2002)
A.
Let A be a square matrix of order n and X be a column matrix of
order n x 1. Now X’ is a row matrix of order 1x n. Hence, X’AX is a matrix of
order 1x1.
Let X’AX = B.Since B is or order 1x1,B’=B and hence,B is
symmetric.
Now, consider,
Example 2. Show that the above matrices are Hermitian.
(M.U.2004)
Solution: We shall show that the last matrix is Hermitian.
The diagonal elements 2, 6, 7 are real.
The elements symmetrically placed with respect to the principal
diagonal are 2 +3i, 2 – 3i, 4 i, 4+ i and 6 +2i, 6 2i.
They are conjugates of each other.
Hence, the matrix is Hermitian.
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Example 3: Show that every square matrix can be uniquely expressed as
the sum of a Hermitian matrix and a Skew-Hermitian matrix.
(M.U. 1999, 2003, 05, 07, 08)
Solution: Let A be a given square matrix.
Now we can write,
say
where
Now,
is
Hermitian
Example 4: Express the matrix
as the sum of a Hermitian and a Skew-Hermitian matrix.
(M.U. 2005)
Solution: We have
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Let
But, we know that P is Hermitian and Q is Skew-Hermitian and A = P + Q
Example 5: Express the following matrix A as P + iQ, where P, Q are
both Hermitian.
(M.U. 2003)
Solution: From A we get
Now let
and let
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If can be verified that both P and Q are Hermitian and that P + iQ
= A.
Example 6: Express the following Skew-Hermitian matrix A as P + iQ,
where P is real Skew-symmetric and Q is real symmetric matrix.
(M.U. 2002, 06)
Solution: We first note that A is a Skew-Hermitian matrix.
Now
Let
and
It can be verified that P is real Skew-symmetric and Q is real
symmetric matrix and A = P + iQ.
Example 7: Prove that the following matrix is orthogonal and hence find
(M.U. 2005)
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Solution:
A is orthogonal and
is the inverse of A.
Example 8: Determine l, m, n and find
is
orthogonal.
Solution: Since for orthogonality
(M.U. 1997, 98, 2000, 02, 03, 09)
.
Also
Example 9: If
State the rank of
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is orthogonal find a, b, c. Also find
(M.U. 2003, 04, 06, 07, 08)
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Solution: Since for orthogonality
Since for orthogonal matrix
the rank of A and hence the
rank of
Example 10:Is the following matrix orthogonal? If not, can it be
cocverted into an orthogonal matrix? If yes, how?
(M.U.2003,04)
Solution: For orthogonality AA’=I.
Since
But the matrix A can be converted into an orthogonal matrix.
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Since,
is the required orthogonal matrix.
Example 11. If (ar,br,cr) where r = 1,2,3 are the direction consines of three
mutally perpendicular lines, then show that
.
(M.U.1996,2002)
Solution:We have
(Using the conditions of perpendicularity of two lines.)
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Example 12. (a): Prove that the matrix
(M.U.1997,2003,06)
Solution:Let us denote the given matrix by A,
Example 13. (a): Show that the matrix
and hence, find A-1 .
is unitary
(M.U.1995,98,2006)
Solution: We have
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Example 14. (a): If N
Awab Sir-89 76 104646
is a
unitary matrix.
(M.U.2005,07,09,11)
Solution: We have
Now,
Let
Hence,
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Example 15: Reduce the following matrix to normal form and find its
rank.
(M.U. 1996, 2003, 09)
Solution: We have
Since the first element in the first row is 1 and the first element in
the second row is 1, we perform
row is 5, we perform
. Since the first element in the third
.
By
Since the first element in the first column is 1 and the first element
in the second column is −1, we perform
first element in the second column is 3, we perform
Since the
. Since the first
element in the fourth column is 6, we perform
By
Since
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is 4 and
is 8, we perform
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By
Since
is 4, we perform
to bring 1 there.
By
Since
we perform
By
∴
∴
which is in normal form.
Rank of A = 3.
Example 16: Find the rank of the matrix by reducing it to normal form.
(M.U.2005,06)
Solution:
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Example 17. Reduce the following matrix to normal form and find its
rank.
(M.U. 1997, 2004)
Solution:
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which is in normal form.
Example 18: Reduce the matrix to normal form and find its rank.
(M.U. 1993, 2003)
Solution:
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Example 19: Reduce the following matrix to normal form and hence find
its rank.
(M.U. 2004)
Solution:
By
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which is in normal form.
Example 20: Reduce the following matrix to normal form and find its
rank.
(MU-2003)
Solution:
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Example 21: Reduce the following matrix to normal form and hence find
its rank.
(M.U. 2012)
Solution:
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.
Example 22: Reduce A to normal form and find its rank where
(M.U.2002)
Solution:
Which is in normal form.
Example 23. Find the value of p for which the following matrix A will
have (i) rank 1,(ii) rank 2,(iii) rank 3.
(M.U.1993,2008)
Solution: The determinant of the matrix A is
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(See Remark 2, page 5-39)
Hence,if
i.e.
Lastly, if p = 3 the matrix (1) becomes
Hence, if p = 3, rank = 1.
Example 24. Find the value of p for which the following matrix A will
have (i) rank 1,(ii) rank 2,(iii) rank 3.
(M.U.2003)
Solution: The determinant of the matrix A is
Hence,
(i)
Hence, the rank of A = 3 if
(ii)
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(iii)
Hence, when p = 2, the rank = 1.
Example 25. Determine the values of p such that rank of
(M.U.2007,09)
Solution: If the rank of A is 3. then
must be zero and at least one minor of
A of order 3 must be non-zero.
Now, the determinant of this matrix is equal to
It can be seen that some minors of order, three are zero but the minor
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Example 26.
PAQ is in normal form.
Awab Sir-89 76 104646
find two matrices P and Q such that
(M.U. 1999,2011)
(Find also the rank of A).
Solution:We first write A = I3 A I3
i.e.
By
By
By
By
By
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(The rank of the matrix is 2).
Example 27: Find non-singular matrices P and Q such that
is reduced to normal form. Also find its rank.
(M.U.2006,07,09)
Solution: We first write
i.e.
=
By
By
By
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Awab Sir-89 76 104646
By
By
By
(5/12) C4
By C34
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Example 28: Find non-singular matrices P and Q, such that A
is reduced
to normal form.
(M.U.2008)
Solution: We first write
By
By
By
By
By
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Example 29: Find non-singular matrices P and Q such that PAQ is in
normal form. Also find the rank of A, and A-1.
(M.U.2002,08,09,10,11)
Solution:
By
By
By
By
By
which is in normal form
To find
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QP
Page 25