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We know AA^{1} = I_{2}
We know
AA^{t} = I_{4}
[AA^{T}]^{1} = [I_{4}]^{1} = I_{4}
ANSWER : c
Solution : Let A be real skew symmetric and suppose λ∈C is an eigenvalue, with (complex) eigenvector v . Then, denoting by H hermitian transposition,
λvHv=vH(λv)=vH(Av)=vH(−AHv)=−(vHAH)v=−(Av)Hv=−(λv)Hv=−λ¯vHv
Since vHv≠0 , as v≠0 , we get
λ=−λ¯
so λ is purely imaginary (which includes 0). Note that the proof works the same for a antihermitian (complex) matrix.
With a completely similar technique you can prove that the eigenvalues of a Hermitian matrix (which includes real symmetric) are real.
The rank of a 3 x 3 matrix C (=AB), found by multiplying a nonzero column matrix A of size 3 x 1 and a nonzero row matrix B of size 1 x 3, is
Then also every minor or order 2 is also zero.
∴ rank (C) = 1
A is a 3 x 4 real matrix and A x = b is an inconsistent system of equations. The highest possible rank of A is
Highest possible rank of A= 2 ,as Ax = b is an inconsistent system.
(P) Singular matrix → Determinant is zero
(Q) Nonsquare matrix → Determinant is not defined
(R) Real symmetric → Eigen values are always real
(S) Ortho gonal → Det erminant is always one
[A] is a square matrix which is neither symmetric nor skewsymmetric and [A]^{T} is its transpose. The sum and difference of these matrices are defined as [S] = [A] + [A]^{T} and [D] = [A] [A]^{T}, respectively. Which of the following statements is True?
∴ Rank (A) = 2
Real matrices [A]_{3 x 1}, [B]_{3 x 3}, [C]_{3 x 5},[D]_{5 x 3}, [E]_{5 x 5} and [F]_{5 x 1} are given. Matrices [B] and [E] are symmetric.
Following statements are made with respect to these matrices.
1. Matrix product [F]^{T} [C]^{T} [B] [C] [F] is a scalar.
2. Matrix product [D]^{T} [F] [D] is always symmetric.
With reference to above statements, which of the following applies?
Consider the matrices X _{(4 × 3)}, Y _{(4 × 3)} and P _{(2 × 3)}. The order or P (X^{T}Y)^{–1}P^{T}] ^{T} will be
When two matrices (x,y) and (a,b) are to be multiplied, the multiplication is possible
when y = a, and the resultant matrix is of the order (x * a)
X^{T} = The matrix in which rows and columns are interchanged
{(2 x 3) x [(3x4) x (4x3)]^{1} x (3x2)}^{T }= { (2x3) x (3x3) x (3x2) }^{T} = { (2x3) x (3x2) }^{T} = 2 x 2
(PQ)^{1}P = Q^{1}P^{1}P = Q^{1}
B^{T} = – B
27 docs150 tests

27 docs150 tests
