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Linear Algebra
3. Use Cayley Hamilton theorem to find the value of the matrix given by
A8 − 5A7 + 7A6 − 3A5 + A4 − 5A3 + 8A2 − 2A + I if the matrix 𝐴 =
[
2 1 1
0 1 0
1 1 2
]
Linear Algebra
(a) Express cos7 θ in terms of multiples of angles.
(b) Express cos4 θ sin3 θ in terms of multiples of angles.
(c) Using complex numbers, prove that the, angles A, B and C of a planar triangle satisfy the relations (i) cos2 A + cos2 B + cos2 C = 1 − 2 cos A cos B cos C
(ii) sin 2A + sin 2B + sin 2C = 4 sin A sin B sin C.
Linear Algebra
Let u=(-1,0,2), v=(3,1,2), w=(1,-2,-2) be vectors in standard position. Compute
1. The orthogonal complement of v
1. The orthogonal complement of v
Linear Algebra
The coefficient matrix of any linear system must be square matrix.
True or false with correct explanation
True or false with correct explanation
Linear Algebra
Suppose A and B are 3x3 matrices. Show that If B is obtained from A by adding 2 times the first row of A to the last row of A, then det(A) =det(B).
Linear Algebra
Suppose A and B are 3x3 matrices. Prove that (i) (A+B)^T = B^T +A^T
(ii) det (AB)^T= det(A).det(B)
(ii) det (AB)^T= det(A).det(B)
Linear Algebra
What is the relationship between the vector (2,1,2) and the line. L defined by the following parametric equations
x=1+2t
y=2+t
z=3-2t
x=1+2t
y=2+t
z=3-2t
Linear Algebra
Suppose A is a 3x3 matrix such that det(A) =16.Find the value of det[(4(A^-1)^T]
Linear Algebra
Suppose C is a 3 x 3 matrix such that det (C) = 4. Find det ( C + C)
Linear Algebra
Find all solutions of this system.
4x + 2y + 3z = 0
3x − y + 2z = 0
x + 2y − z = 0
Compare the value of the determinant of the coefficient matrix to zero using the nature
of the solution.
