Linear Algebra Answers

Find the inverse of the matrix
1 −1 0
2 −1 1
1 1 −1

using row reduction.

Let B1 ={(1,1),(1,2)} and B2 ={(1,0),(2,1)}. Find the matrix of the change of basis from B1 to B2

Which of the following matrices are Hermitian and which are Unitary? Justify your answer.


A = 1 i 0 B= 1/√2 0 -1/√2
-i 1 1-i 0 1 0
0 1+i 2 V2i 0 V2i

Use Gram-Schmidt orthogonalisation process to find an orthonormal basis for the subspace of C^4 generated by the vectors (1,i,0,−i), (−i,0,1,2) and (0,−i,1,1).

Derive the matrix of translation in 2-dimensional plane? Why its first 2 column is identity matrix

Check whether the forms
2x^2+3y^2+5z^2−4xz−6yz

and 4x^2+3y^2+z^2−6xy−2xz

are orthogonally equivalent

Q. Derive the translation matrix? Why we take its first 3 column identify matrix.

A tank is fitted with two taps A and B of different size. If both the taps are opened simultaneously it takes 4 hours to fill the tank. If only tap A is opened for 1 hour and then only tap is opened for 4 hours the tank becomes half. Find the time taken by tap B alone to fill the tank.

A 6 hours

B 8 hours

C 12 hours

D 14 hours

Consider the following system of equations:
x1−3x2−x3 = 3
x1+5x2+3x3 = 1
−x1+7x2+3x3 = 1
Check whether the system of equations have a solution or not

Let
A =5 4 −4
6 7 −6
12 12 −11

a) Find the adjoint of A. Find the inverse of A from the adjoint of A.
b) Find the characteristic and minimal polynomials of A. Hence find its eigenvalues and eigenvectors.
c) Why is A diagonalisable? Find a matrix P such that P^(−1) AP is diagonal.
d) Verify Cayley-Hamilton theorem for A. Hence, find the inverse of A.