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Matrices Some special types of Matrices Some special type of matrices
Symmetric & Skew symmetric
Symmetric matrix: If then square matrix
An is called symmetric matrix. ∀ i.e. aij = aji i & j Skew symmetric matrix
If then square matrix An is called skew symmetric matrix. ∀ i.e. aij = - aji i & j ∀ Clearly aii = 0 i Q. If A is a square matrix then prove that:
(a) is a symmetric matrix
(b) is a skew symmetric matrix. Q. If A is a square matrix then prove that: (a) is a symmetric matrix Q. If A is a square matrix then prove that: (b) is a skew symmetric matrix. Remark: Every square matrix A can be represented as a sum of symmetric & skew symmetric matrix. Q. If A & B are symmetric matrices of same order then prove that AB – BA is skew symmetric matrix. Q. If A & B are symmetric matrices of same order then prove that AB – BA is skew symmetric matrix.
Solution: (AB - BA)’ = (AB)’ - (BA)’ = B’A’ - A’B’ = BA - AB = - (AB - BA) Q. If A is symmetric (skew symmetric) matrix then prove that BTAB is also symmetric (skew symmetric) matrix. Q. If A is symmetric (skew symmetric) matrix then prove that BTAB is also symmetric (skew symmetric) matrix.
Solution: Also, If A is skew symmetric Given, A is symmetric AT = −A Thus AT = A Again Now, we have to check for (BTAB) (BTAB)T = BTAT(BT)T Taking Transpose = BT(–A)B T T T T T T (B AB) = B A (B ) = –(BTAB) = BTAB Thus, it is skew symmetric matrix Thus it’s a symmetric matrix Multiple correct JEE Advanced 2015
Q. Let X3×3 & Y3×3 are non zero skew symmetric matrices & Z3×3 is a non-zero symmetric matrix then which of the following are skew symmetric.
3 4 4 3 A Y Z – Z Y
44 44 B X + Y
4 3 3 4 C X Z – Z X
23 23 D X + Y
Q. Let X3×3 & Y3×3 are non zero skew symmetric matrices & Z3×3 is a non-zero symmetric matrix then which of the following are skew symmetric.
3 4 4 3 A Y Z – Z Y
44 44 B X + Y
4 3 3 4 C X Z – Z X
23 23 D X + Y
Q. Let X3×3 & Y3×3 are non zero skew symmetric matrices & Z3×3 is a non-zero symmetric matrix then which of the following are skew symmetric. Solution: = –(X4Z3–Z3X4) Given X and Y are skew symmetric Also, in option (d) Then, XT = −X and YT = −Y (X23 + Y23)T = – X23 – Y23 Also, ZT = Z = – (X23 + Y23) Now, in (c) option, (X4Z3 – Z3 X4)T = (X4Z3)T –(Z3X4)T = (ZT)3 (XT)4 – (XT)4 (ZT)3
3 4 4 3 = Z X – X Z Thus both are skew symmetric. SAWAAL Q. Let A be the set of all 3 x 3 matrices which are symmetric with entries 0 or 1. If there are five 1’s and four 0’s, then number of matrices in A is
A 6
B 12
9C 9
D 18
Note: Solution at the End Note:
1. Determinant of skew symmetric matrix of odd order is zero. 2. Determinant of skew symmetric matrix of even order is a perfect square. Hermitian & skew Hermitian matrix
Transposed Conjugate Transpose of conjugate of a matrix is called its transposed
Ө conjugate. (Denoted by or A
If then
& Hermitian Matrix:
A Square matrix is called Hermitian matrix if
i.e. Clearly, & hence diagonal elements will be purely real. Ex: Skew Hermitian matrix:
A square matrix is called skew Hermitian if
i.e.
Clearly, & hence diagonal elements will be purely imaginary.
A square matrix is called orthogonal if AAT = I
Remark:
If A is an orthogonal matrix then its determinant must be ±1.
IIT 2003
Q. If is orthogonal, where a, b, c are positive &
abc = 1 then find value
A 2
B 3
C 4
D None of these
Q. If is orthogonal, where a, b, c are positive &
abc = 1 then find value
A 2
B 3
C 4
D None of these
Q. If is orthogonal, where a, b, c are positive &
abc = 1 then find value Solution:
If a, b, c are +ve then If A is an orthogonal matrix then its determinant must be ±1. Observe:If is orthogonal then:
If is orthogonal then:
Equating values we get:. 1. All the rows are unit vector. 2. Any two rows are perpendicular vectors. Similarly, we can show same results for columnS 2. Any two rows are perpendicular vectors. Try to observe directly that following are orthogonal matrices.
(3) (1)
(2)
Q. If = 0 & = 1 then value of
A 0
B 1
C -1
D ±1
Q. If = 0 & = 1 then value of
A 0
B 1
C -1
D ±1 Solution:
Let
Now
Thus A is orthogonal matrix Thus |A| = ±1 Q. If is a matrix such that where
I is 3×3 identity matrix, then ordered pair (c, b) is
A (-2, -1)
B (-2, 1)
C (2, 1)
(-2, -1) Hint. D Try by checking the options. Q. If is a matrix such that where
I is 3×3 identity matrix, then ordered pair (c, b) is
A (-2, -1)
B (-2, 1)
C (2, 1)
D (-2, -1) Solution: Let’s put a = − 2 and b = − 1. Then,
Thus (−2, −1) IIT 2005
Q. Find , where , & Q = PAPT Solution:
Idempotent Matrix
A square matrix is called idempotent if A2 = A Clearly, An will also be equal to A Q. If AB = A & BA = B then show that A, B, AT & BT will be idempotent. Q. If AB = A & BA = B then show that A, B, AT & BT will be idempotent.
Solution: Given, AB = A B(AB) = BA So A is idempotent. BB = B Now A2 = A B2 = B (A2)T = AT So B is idempotent (AT)2 = AT Similarly Thus AT is idempotent. BA = B (AB)A = AB AA = A ⇒ A2 = A Note:
(I + A)n can be expanded like Binomial theorem: Q. If A is an idempotent matrix then show that:
Hint:
I and A commute with each other: I × A = A × I = A Q. If A is an idempotent matrix then show that:
Solution:
Given, A is idempotent matrix. Then
Q. If A is an idempotent matrix then
A
B
C Hint: D
Q. If A is an idempotent matrix then
A
B
C
D
Q. If A is an idempotent matrix then
Solution: Given, A is idempotent matrix Then, Since for Involutory Matrix
A square matrix is called involutory if A2 = I. Nilpotent Matrix
A square matrix is called nilpotent matrix of order m if:
&
Note: There is no way to check whether matrix is nilpotent or not, other than checking the powers manually.
Q. If A is an involuntary matrix, then (A – I)3 + (A + I)3 – 7A is equal to
A A
B I - A
C I + A
D 3A
Q. If A is an involuntary matrix, then (A – I)3 + (A + I)3 – 7A is equal to
A A
B I - A
C I + A
D 3A
Q. If A is an involuntary matrix, then (A – I)3 + (A + I)3 – 7A is equal to
Solution: Singular & Non-singular Matrix
A matrix is called singular if its determinant is zero, otherwise it is called non-singular. In the last Few Days/Month you need
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8th Jan 2020-(Shift 1) Q. Let A be the set of all 3 x 3 matrices which are symmetric with entries 0 or 1. If there are five 1’s and four 0’s, then number of matrices in A is
A 6
B 12
9C 9
D 18 Solution: