Linear Algebra Answers

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.

Use the properties of determinants to evaluate the following determinant:

(b+c)^2 a^2 a^2
b^2 (c+a^)2 b^2
c^2 c^2 (a+b)^2

a) Check whether the forms 2x^2+3y^2+5z^2−4xz−6yz and 4x^2+3y^2+z^2−6xy−2xz are orthogonally equivalent.
b) 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).
c) 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 √2i 0 √2i

a) Let a quadratic form have the expression x^2+y^2+2z^2+2xy+3xz with respect to the standard basis B1 ={(1,0,0),(0,1,0),(0,0,1)}.
Find its expression with respect to the basis B2 ={(1,1,1),(0,1,0),(0,1,1)}


b) Consider the quadratic form Q:2x^2−4xy+y^2+4xz+3z^2
i) Find a symmetric matrix A such that Q = XtAX.
ii) Find the orthogonal canonical reduction of the quadratic form.
iii) Find the principal axes of the form.
iv) Find the rank and signature of the form

The roots of the following equation are 5.8210,1.6872 and -5090. Using Muller method approximates the root 5.8210 upto four decimal places.
f(x)=xcube-7xsquare+6x+5

Using Newton-Raphson method approximates the root of the following equation in the interval (0,1) upto the three decimal places.
f(x)=3x-cosx-1
Note: All the calculations should be in radian.

hello,
Please help me solve this

Consider for every real number a the linear system of equations:

{ x +( a + 1 )y + a^2 z = a^3
{ ( 1 - a )x +( 1 - 2a )y = a^3
{ x +( a + 1 )y + az = a^2


a)Find the solution for a = 2

b)Find the values of a for which the system has no solution, infinitely many solutions, and a unique solution

c)Find the solution for a = -1

Compute the determinant of :
∣ ∣ ∣ −1 −4 −1 −2 ∣ ∣ ∣

Compute the determinant using elements in the first row:
A∣1 5 4∣
∣0 −7 −8∣
∣3 7 1∣

a. -7
b.32
c. -27
d. 3

Compute the determinant of :
∣−1−4∣
∣−1−2∣

a. 8
b. 4
c. 3
d. -2