Which ofthe following is the LU-factorization of the matrix a) b) 1. c) -1 [1 01 0] d) -3

Q: Let A be an m x n matrix with m < n. Which of the following statements is always false?

A:

Q: A, 2x2 tipinde tekil(singüler) bir matris olmak üzere A.Ek(A) matrisi aşağıdakilerden hangisidir? A…

A: We're going to solve

Q: Let A be a m×n matrix. Prove that AB and Ba both exist only if B is an n×m matrix.

A: Explanation of the answer is as follows

Q: if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))

A:

Q: Find the change-of-coordinates matrix from B to the standard basis in R. 9 400 B = 3 -2 3 -6 5 PB =

A: Given that B=933,-20-6,4-25

Q: For the matrix A in , use the Cayley ­Hamilton Theorem to compute A2, A3, and A4 by expressing each…

A: Given matrix is

Q: Let A be a 4x6 matrix. what is the dimension of Col A if the nullity is 2? Explain

A: 4

Q: Which of the following is the LU-factorization of the matrix a) 1 b) 0 1 1 c) d)

A:

Q: let Q(x)= x12 -8x1x2 -5x22 , then matrix A of the quadratic from is A=?

A: Explanation is given below..

Q: which of the following is the cofactor A13 of matrix?

A: Simply omit that raw and column whose co-factor matrix required and write rest all elements in same…

Q: State Cayley-Hamilton theorem and prove this for following matrix [1 1 A 1 2 -1 %3D [5 8

A:

Q: Suppose that A and B are matrices having n rows. Prove that M(A|B) = (MA|MB) for any m ×n matrix M.

A:

Q: a) Factor the matrix A into the product XDX- where D is diagonal. (-1 -2 2 A = 4 3 -4 -2 b) Verify…

A:

Q: Find diagonal matrix D is similar with A a A =[? -5 And make sure can transtor diagomalizablewith

A:

Q: Let A be an nxn matrix. Prove that dim(col(A²)) < dim(col(A)).

A: Given: Assume that, A is an n×n matrix.

Q: Q/ Let A: am et fox)= 2x- 子 4x+5 whene I Find FCA) is unit matrix

A:

Q: b) Prove: Let A be an m × n matrix, R" = NS(A) RS(A).

A: Given: Let A be an m×n matrix , Rn=NSA⊕RSA

Q: Use the Cayley-Hamilton theorem to find A1, A°, and A“ for the given matrix A. 10 -8 - 4

A:

Q: Use A-1 =(1/detA)adj A to compute the in­verse of the coefficient matrix x + y = 1   x-y=2

A: x+y=1 x-y=2 We need to find the inverse using A-1 =(1/detA)adj A

Q: b) Prove: Let A be an m x n matrix, R" = NS(A) + RS(A). %3D

A: A be an m×n matrix, we have to prove that ℝn=NSA⊕RSA  …1.

Q: Г1 3 2 5 1 [3 4 2

A:

Q: Find the elementary matrix Ej such that EA = B where 7 A =4 -11 -1 and B = 4 -11 -1 5 6 5 1 12 1 12…

A: We have to solve given problem:

Q: Find the LU tactorization of 2 -4' -4 7 10 18 That is, write A = LU where L is a lower triangular…

A: I have used Gauss elimination method to find LU decomposition

Q: 2rị + 13 T: R' → R?, T 21, 22, 13, L4 ER -12 - 3r4

A: Given  and the standard bases are given by

Q: Let A be a square matrix such that A2 = 0. Prove that (Af)2 = 0

A: Given that A is a square matrix and A2=0. In order to prove (At)2=0 following theorem are used.…

Q: Let A be a 3 x 3 matrix such that 3a(1)- 2a(2)+7a(3)=0. Is A nonsingular? Explain.

A: A is singular matrix if rank is not full

Q: Let A = CD, where C and D are n × n matrices and C is invertible. Prove that the matrix product DC…

A: A = CD, where C and D are n × n matrices, and C is invertible. We need to Prove that the matrix…

Q: Let A be an m×n matrix, X an n×r matrix, and B an m × r matrix. Show that AX = B if and only if Axj…

A:

Q: In exercises 8 to 13 , explain how your calculations justify your answer, and mention an appropriate…

A:

Q: Let A = [aij] be a square matrix If B is obtained by adding a multiple of one row (column) of A to…

A:

Q: Find the change-of-coordinates matrix from B to the standard basis in Rr. 8 B=. 4 -8 7 3 7 Pe =0

A:

Q: Let M be a real square matrix such that M3 + M = 2M². Prove that M is diagonalisable.

A:

Q: 16. Prove that for any square matrix A, ||A||ž E ||A* A||2.

A: Given: A is a square matrix.

Q: Explain why the columns of an nxn matrix A span R" when A is invertible. Choose the correct answer…

A:

Q: Determine the matrix C such that (CT – 21) 1 1 2 2 3

A:

Q: Let A be an m × n matrix. Show that ATA and AAT are both symmetric.

A: Given A be an m×n matrix. we have to show that ATA and AAT are both symmetric.we will make use of…

Q: Prove the following: A(BC) = (AB)C for all 2x2 matrixes

A: Consider the matrices: A=a1a2a3a4B=b1b2b3b4C=c1c2c3c4

Q: Find the change-of-coordinates matrix from B to the standard basis in R". 6. -4 2 B= 5 -4 7. -3 7…

A: Solution

Q: Consider the ordered bases B = {1+x, 1 – x} and B' = {u1, u2} for P1(IR), and suppose that - 2 Q =…

A: Given that  β={1+x, 1-x} and β'={u1,u2} are order bases for P1(ℝ) and            Q=2-145    is the…

Q: Find the change-of-coordinates matrix from B to the standard basis in R". 8 3 B= 7 -3 5 - 4 8 PB II

A:

Q: Verify that P = is an orthogonal matrix. 1/32/3 N3

A: Given the matrix P=23-23132313-23132323We need to check whether the given matrix is orthogonal…

Q: Let A = [aij] be a square matrix. a. If A has a zero row (column), then prove that det A = 0.

A: Let A = [aij] be a square matrix and If A has a zero row (column). We need to prove that det A = 0.…

Q: Find matrix B such that AB = C. Hint: B = A^-1 x C *

A:

Q: 20. Can every vector in R* be written as a linear combination of the columns of the matrix B above?…

A:

Q: Let B = {b1, b2} and C = {c1, c2} be two bases for R, where %3D bị b2 C1 = C2 = 8 b1 what is the…

A:

Q: Show that for matrices A, B E RXn, a vector w = ABx can be written as w = E,(b{ x)a;, %3D :D1 where…

A: Since, A,B∈ℝn×n. ∴ A, B have 'n' rows and 'n' columns with entries as real numbers. The given vector…