(2 points) Let (2 points) Let A=[% ] () Write A as a product of 4...
Let A = [111] 1 2 3. Write A as the product of elementary matrices. (1 4 5
2. (15 pts; 8,7) Let (a) Find the inverse of the matrix X. (b) Write X-1 as a product of elementary matrices. (You only need to give the list of elementary matrices in the right order. There is no need to multiply them out. )
a) b) hufi 2 1 Wait (3 marks) Let A = 1. Write A-1 as a product of elementary matrices. Show your work. 1 -1] [ 1 0 1] - (3 marks) Let A = -14 2 1 . Determine all values of q for which A is nonsingular. IO 1 q
3. Let A 2 -30 1 0 -2 2 0 (i) Compute the determinant of A using the cofactor expansion technique along (a) row 1 and (b) column 3. (ii) In trying to find the inverse of A, applying four elementary row operations reduces the aug- mented matrix [A1] to -2 0 0 0 0 -2 2 1 3 0 1 0 1 0 -2 Continue with row reductions to obtain the augmented matrix [1|A-') and thus give the in-...
1. Let A and B be two 4 by 4 matrices with (let A =-2 and det B-1-8. Find det(-2.1' B) 2. Assume that A is a 4 x 4 matrix and det (Adj(A))-8, find det(A) 3. Find the inverse the given matrice by way of elementary row operations
2-1 1 Write M1 0as a product of elementary matrices and find the inverse of M.
HW10P5 (10 points) Let A be the matrix A =13 5 0 (3 pts) Find the elementary matrices that perform the following row operations in sequence: a. 21 * 2 2. E31 : R3 R1R3 b. (3 pts) Show that the elementary matrices you found in (a) can be used as elimination matrices to determine the upper triangular, U, matrix of A. (4 pts) Find the lower triangular, L, matrix that verifies A C. = LU.
[M2] Let -1] 2 A = 2 1 -2 3 (a) Find A-1, (b) Use the inverse matrix above to solve the system -2x1 + 2x2 – x3 2, X1 + x2 + 2x3 = -1, 2^1 — 2л2 + 3х3 — 5. (c) Write the following matrix A as a product of elementary matrices. |0 A = |1 -2 0 3 5
4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 1 1 1 0 This is consisting of upper-triangular matrices. Let B= a basis for V. (You do not need to prove this.) (a) (8 points) Use the Gram-Schmidt procedure on 3 to find an orthonormal basis for V. Find projy (B) (b) (4 points) Let B= 4. Consider R2x2 with inner product (A, B) tr(ATB), and let V CR2x2 be the subspace 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