Linear Algebra - II Quiz Test

The eigen values of A and its transpose A^T are .................

• Same
• Unrelated.
• Opposite.
• Different.

The characteristic roots of a Hermitian matrix are all ........................

• complex.

• imaginary.
• either purely imaginary or zero

• real

The characteristic roots of a real symmetric matrix are all ........................

• complex
• purely imaginary or zero

• rational
• real

The characteristic roots of a skew Hermitian matrix are ........................

• purely imaginary or zero

• positive

• complex

• real

The characteristic roots of a real skew symmetric matrix are ........................

• zero

• real

• positive

• pure imaginary or zero

Zero is an eigen value of A if and only if A is a ........................ matrix.

• Zero

• Identity

• singular

• orthogonal

Find the eigen values of A⁵ when A =

• 3,4,1

• 3⁵,4⁵,1⁵

• 2⁵,6⁵,1⁵

• 2,6,1

Find the matrix of the bilinear form f(x,y) =x₁y₂-x₂y₁ wit respect to the standard basis in V₂(R)

• [[1, 1], [-1, -1]]
• [[0, -1], [1, 0]]
• [[1, 0], [0, -1]]
• [[0, 1], [-1, 0]]

Find the matrix of the bilinear form f(x,y) = x₁y₁+x₃y₃ with respect to the standard basis V₃(R)

• [[1, 0, 0], [0, 1, 0], [0, 0, 0]]
• [[1, 0, 0], [0, 0, 0], [0, 0, 1]]
• [[0, 1, 0], [0, 0, 0], [0, 0, 1]]
• [[0, 0, 1], [0, 0, 0], [0, 1, 0]]

L(V, V, F) is a vector space of dimension .........................

• 2n

• n²

• n

• n/2