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Comprehension:
Direction: Based on the following information, answer the questions:
Abhi, Badri, and Chintan were given the task of creating a square matrix of order 3.
Below are the matrices created by them. A, B, and C are the matrices created by Abhi, Badri, and Chintan respectively.
Let matrix B = P + Q, where P is a symmetric matrix and Q is a skew  symmetric matrix. What is Q?
For any matrix A, (A + A’) is always symmetric and (A  A') is always skewsymmetric.
where P = 1/2(A+ A'), a symmetric matrix and Q = 1/212(A  A'), is a skewsymmetric matrix.
Caluclation:
∴ Required symmetric matrix Q
= 1/2 (BB')
which is the required answer.
Direction: Based on the following information, answer the questions:
Abhi, Badri, and Chintan were given the task of creating a square matrix of order 3.
Below are the matrices created by them. A, B, and C are the matrices created by Abhi, Badri, and Chintan respectively.
What is (A^{T})^{T}
If A is any matrix, the (A^{T})^{T} = A.
We have,
∴ (A^{T})^{T}
= A
Given:
A is a symmetric matrix,
⇒ AT = A or a_{ij} = a_{ji}
So, by property of symmetric matrices
lf the order of A is 4 × 3, the order of B is 4 × 5 and the order of C is 7 × 3, then the order of (A^{T}B)^{T} C^{T} is
Given:
Order of A is 4 × 3, the order of B is 4 × 5 and the order of C is 7 x 3
The transpose of the matrix obtained by interchanging the rows and columns of the original matrix.
So, order of AT is 3 × 4 and order of CT is 3 × 7
Now,
A^{T}B = {3 x 4} {4 × 5} = 3 x 5
⇒ Order of ATB is 3 x 5
Hence order of (A^{T}B)^{T} is 5 x 3
Now order of (A^{T}B)^{T} C^{T} = {5 x 3} {3 x 7} = 5 x 7
∴ Order of (A^{T}B)^{T} C^{T} is 5 x 7
⇒ [(2x  9)x + 8 x 4x] = 0
⇒ [2x^{2}  9x + 32x] = 0
⇒ 2x^{2} + 23x = 0
⇒ x(2x + 23) = 0
⇒ x = 0 or  23/2
If A is an Involuntary matrix and I is a unit matrix of same order, then (I − A) (I + A) is
Given that A is involuntary matrix,
⇒ A^{2} = I
Now,
(I − A) (I + A) = I^{2} – IA + AI − A^{2}
⇒ I – A + A – I (∵ A^{2} = I)
⇒ 0
∴ (I − A) (I + A) is zero matrix.
Consider the following question and decide which of the statements is sufficient to answer the question.
Find the value of n, if
Statements∶
1. AB = A
2.
From statement 1∶
AB = A
We cannot find anything from this statement.
From statement 2∶
We cannot find anything from this statement.
Combining statement 1 and 2∶
∴ We cannot find the value of n from both statements together.
Given: A^{2}  2A  I = 0
⇒ A.A  2A = I
Post multiply by A^{1}, we get
⇒ AAA^{1}  2AA^{1} = IA^{1}
⇒ AI  2I = A^{1} [∵ AA ^{1} = A ^{ 1}A = I]
∴ A^{1} = A  2
the inverse of A is A  2
Now,
Hence Option 1st is correct answer.
Given
47 videos119 docs75 tests

47 videos119 docs75 tests
