Prove that type 1 elementary matrix is a product of type 2 and 3 elementary matrices
Prove that type 1 elementary matrix is a product of type 2 and 3 elementary matrices
1 3 1. Find the inverse using elementary matrices A 2-3 Find a sequence of elementary matrices whose product is the given matrix. 2-H 4 3 Find an LU-factorization. 3 01 6 1 1 3. -3 1 3 1. Find the inverse using elementary matrices A 2-3 Find a sequence of elementary matrices whose product is the given matrix. 2-H 4 3 Find an LU-factorization. 3 01 6 1 1 3. -3 1
2. Write the product of a sequence of elementary matrices which equals the given non-singular matrix: [ 11 2 3 3. Given the matrix A = 01 - write the matrix of minors of A, the matrix of cofactors of A, the adjoint 12 2 2 matrix of A, and use the adjoint of A, to write the inverse of A. 4. Determine whether the set of vectors is linearly dependent or linearly independent. Justify your answer. 13
Let A = [111] 1 2 3. Write A as the product of elementary matrices. (1 4 5
We have discussed elementary elimination matrices M_k in class. Prove the following two properties of elementary elimination matrices, which are very important for making LU factorization work efficiently in practice: M_k is nonsingular. Represent M_k^-1 explicitly and show that M_kM_k^-1 = M_k^-1M_k = I. The product of two elementary elimination matrices M_k and M_j with k notequalto j is essentially their "union"; and therefore they can be multiplied without any computational cost.
9. s: ['o] Write s' as the product of elementary matrices, that is, matrices from the identity matrix through a single row are obtained that operation
Show that is invertible by representing A as a product of elementary matrices. Then, represent A^-1 as a product of elementary matrices ГО 0 5 tA = 0 1 0 1 -3 00
2. Consider matrix A = 5 0 1 2 Find elementary matrices E1, E2 and E3 such that E3E2E1A=I.
QUESTION 6 a) Prove the product of 2 2 x 2 symmetric matrices A and B is a symmetric matrix if and only if AB=BA. b) Prove the product of 2 nx n symmetric matrices A and B is a symmetric matrix if and only if AB=BA.
[1 2 2 3 2 3 (a) Factorize the matrix A =| 2 | into elementary matrices. (b) Write the condition for positive and negative definite quadratic forms. Reduce the quadratic form q=4x7+3x2-x?+2x2x3-4x3x1+4x1x2 to the canonical form. Hence find rank, index and signature of q. Write down the corresponding equations of transformation.
2-1 1 Write M1 0as a product of elementary matrices and find the inverse of M.