(a) Prove that if A and B are similar n x 11 matrices, then tr(A) = tr(B).  (b ) How would you define the trace of a linear operator on a finite dimensional vector space? Justify that your definition is well- defined.

Q: prove that if A and B are invertible matrices with the same size then AB is also invertible and :…

A: A and B are invertible. So, AB=BA=I

Q: 4. Let 1 A = 8 -1 -1 -4 1 -1 Prove that A is not diagonalizable over R, but is diagonaliz- able over…

A:

Q: 1 d) The set of all 2 × 2 matrices of the form 0 1 a (a E R) with usual matrix addition and scalar…

A:

Q: 3. Any two scalar 5 x 5 matrices are linearily dependent. 4. For any two vectors u, v in a vector…

A:

Q: Let M2x2 be the vector space of real 2 x 2 matrices with entrywise addition and scalar…

A: If 0 is an eigen value of matrix A then it is singular.  Since det(A) is equal to product of eigen…

Q: Find a QR factorization of the matrix in Exercise 12.

A: Given matrix is A=135-1-31023152158 To determine the QR factorization of the given matrix

Q: The matrices A = A2 = Az A4 = form a basis for the linear space V = R²×2. Write the matrix of the…

A:

Q: Prove that the rank of a matrix, every element of which is unity is 1

A: Rank of a matrix refers to the number of linearly independent rows or columns of the matrix.To find…

Q: Let [1 0.25] A : 1 Find an invertible matrix X and a diagonal matrix D such that X-'AX = D.

A: If for matrix A,      X-1 AX =D  Then matrix A is Diagonalisable,   And where D is diagonal matrix…

Q: Question 1: Let V (Set of all invertible N x N matrices} with the standard definition of matrix…

A: If a matrix is invertible then the inversion of elements of the matrices will not change the…

Q: Which of the following is true or false? a) If  Q and  P  are the orthogonal matrices then  the…

A: As per guidelines, we are supposed to solve only first subdivision. Please repost other question…

Q: Give an example of an operator whose matrix with respect to some basis contains only O's on the…

A:

Q: Consider A and B are two square matrices and B is a scalar. Calculate (BA^T B)^(-1) = *

A:

Q: аxis 12 Write the standard matrix for the transformation T: R2 → R2 that projects a vector onto the

A: Answer

Q: -2 0 2 ) Let A = Find an invertible matrix U such that UA = R %3D 1 2 3 is in reduced row-echelon…

A: We will use elementary row operation to get U.

Q: 8. Let A be a matrix 4 0 0 0 0 4 0 0 0 0 2 0 1 0 0 2 Determine if A is diagonalizable, and if…

A: Matrix A is diagonalizable if there exists an invertible matrix P and a diagonal matrix D, such that…

Q: TRUE/FALSE If A is nonsingular, then A can be factored into a product of elementary matrices.

A: Given: The statement is “ If A is non-singular, then A can be factored into a product of elementary…

Q: Good morning, could you help me with the following demonstration? Thank you very much Prove that 4.…

A:

Q: 4) Confirm the results of the Fundamental Theorem of Linear Algebra for each of the matrices A and B…

A:

Q: a) A is an orthogonal matrix b) B is a singular matrix

A: (a) Orthogonal matrix are square matrices which , when multiplied with its transpose matrix results…

Q: If A is a n x n matrix with NON zero determinant, then its columns are a basis for a n dimensional…

A: “Since you have asked multiple questions, we will solve the first question for you. If youwant any…

Q: 3. What is the dimension of the vector space consisting of five-by-one column matrices where the…

A: Option B i.e 4

Q: False + 1. If A and B are square matrices of the same size with B invertible, then det(B¯'AB) =…

A: Since we are given with the multiple number of question and our guideline is solve first question…

Q: Show that the matrix 2 1 A = 1 3 2 1 1 1 is invertible, but that A cannot be written as the product…

A: Solve the given problem.

Q: If  det(A) =|A|= 0, then A is called a non-singular matrix  True  False

A: Given statement is false because

Q: True or false? det(A) is defined only for a square matrix A.

A: True/False det(A) is defined only for a square matrix A

Q: If A is an n x n diagonalizable, constant matrix, then the set of solutions to x'(t) = Ax(t) is a…

A: Remark: An eigenspace of a square matrix A is the collection of eigenvectors associated with each…

Q: 5) Give an example of an operator whose matrix with respect to some basis contains only nonzero…

A: We have to give an example of an operator whose matrix with respect to some basis contains only…

Q: 13. a) Show that if A and B are n x n matrices, at least one of which is invertible, then AB and BA…

A: (a) If A and B are two n×n matrices, and one of them is invertible, then show that AB and BA are…

Q: 1. Let V be the set of all 2 x 2 matrices A= [a b such that the product abcd = 0 with standard…

A: As per our guidelines we are supposed to solve only first three subparts of the multipart problem.…

Q: Determine whether the matrix is orthogonal. An invertible square matrix A is orthogonal when A−1 =…

A: Now transpose of this matrix is,

Q: ab ={" E M2va a =- d is a vector space under standard matrix addition and scalar multiplication. خطأ…

A: We have to check

Q: Prove that the inverse of a Hermitian matrix is also Hermitian (transpose s-1 S = I).

A:

Q: If v is an eigenvector of A, then ū is an eigenvector of A".

A:

Q: b) Evaluate det[(2A)-1 (3A")], where A is any invertible (2 x 2) matrix.

A:

Q: Determine whether the matrix is orthogonal. An invertible square matrix A is orthogonal when A−1 =…

A:

Q: 1. Let V be the set of all 2 x 2 matrices 4- such that the product abcd =0 with standard operations…

A: According to the Bartleby guidelines, Solve only first three parts Let V be the set of all 2×2…

Q: 1) Suppose that T is a linear operator with standard matrix 8 3 6 0 2 0 0 7 Is T one-to-one?…

A: Given the linear operator T:ℝ3→ℝ3 836025007 Now, we find determent of T.…

Q: da basis for the null space of matrix A. -2 5 0 1

A:

Q: If a vector is in the range of a matrix A, then it is also in the null space of A. true of false?

A: Answer

Q: An n x n matrix A is invertible if and only if it is row equivalent to I. O True False

A: Given statement is "An n×n matrix A is invertible if and only if it is row equivalent to In."

Q: Generalize the result by proving that, for any comfortable non-singular matrices A,B and C the…

A: Consider the given information. ABC-1=C-1B-1A-1 Multiply by pre and post C.…

Q: Prove that if a symmetric matrix is invertible, then its inverse is symmetric also

A: Let A be a invertible symmetric matrix. To prove that, Inverse of a symmetric matrix is symmetric.…

Q: Let V be a finite-dimensional vector space and T : V V be a linear transformation. If B and C are…

A: use property that similar matrix have same eigen value

Q: 3. Determine whether A is diagonalizable and, if so, find an invertible matrix P and a diagonal…

A: Hi, since you have asked multiple questions, we will solve the first question for you. If want any…

Q: If an upper triangular matrix A is singular then at least one diagonal element is zero. O True O…

A: We have to find out

Q: Let A be a 4x4 singular matrix. What is the smallest possible dimension of Null Space of A? A) 2

A: Using,  Rank- Nullity Thereom

Q: Question 1 Consider the set of matrices -{(: :) -ahcer} a b V = : а, b, с€R || Is this set V a…

A: The given set of matrices is                                  V=abbc : a, b, c∈ℝ

Q: (i) a rotation by an angle of π/4 clockwise, (ii) an orthogonal projection onto the vector u = (1,…

A: The vector or equation R2 is not given

Q: a. Prove that if A and B are n xn non-singular matrices, then (AB)-1 = B-'A-1.

A: If A and B are n x n non-singular matrices then prove that (AB)-1=B-1A^-1

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.