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Suppose T is invertible then show that (T-1)* = (T*)-1
Show that a triangular matrix is normal if it is diagonal
The minimal polynomial of a n×n matrix is of degree n.
True or false with full explanation.
If A is a unitary matrix, then all its eigen values are 1.
True or false with full explanation
The sum of two invertible matrices is an invertible matrix.
True or false.
Reduce the quadratic form 5x^2-4xy+8y^2 to its orthogonal canonical form, clearly giving the transformations being used. Also give a rough sketch of the curve representing this canonical form.
Using the Gram-Schmidt prodecure find an orthornornal set of vectors corresponding to the ordered basis B {(1,1,1),(1,1,0),(1,0,0)} of R^3. Also find a basis dual to B.
Let {u,v,w} be an orthornornal set of vectors in R^3. Show that they are linearly independent over R. Check whether u-v, u+v,w are orthogonal over R or not.
Let T: R^3→R^2 be given by:
T(x2,x2,x3)= (X1+x2+x3,x2+x3)
Prove that T is a linear transformation. Also find the rank and nullity of T.
Check whether or not the matrix
A=[1 2 3]
[0 2 3]
[0 0 3]
Is diagonalisable. If it is find a matrix P so that P^-1AP is a diagonal matrix. If A is not diagonalisable, obtain its minimal polynomial.
