Show that if A is skew symmetric, then the elements on the main diagonal of A are all zero.

Q: If A ∈ R^n×n is invertible, then columns of A^−1 are linearly independent. Explain why.

A:

Q: If A is diagonalizable and B is diagonalizable, must it be true that AB is diagonalizable? Give a…

A: If A is diagonalizable and B is diagonalizable then it is not necessary for AB to be diagonalizable.…

Q: Show that matsix is not 15 diagonalizable A- 2 -3 2.

A:

Q: Prove that if the rows of A are linearly dependent, then so are the rows of AB.

A: Let us assume that the product AB makes sense. To prove that is if the rows of A are linearly…

Q: Show that if A is any m × n matrix, then Im A = A and AIn = A.

A: We know that if A is a matrix of order m x n and B is another matrix of order n x p, then their…

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

A:

Q: Find two examples to demonstrate that if (a-d)2+4bc =0 then A may or may be diagonalizable

A:

Q: Prove that if A is an n × n matrix, then A − AT is skew-symmetric.

A: Let  A be an n × n matrix, To prove : A-AT is skew-symmetric, that is (A-AT)T=-(A − AT)

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: Prove that if the two columns of A ∈ M2×2(F) are identical, then det(A) = 0

A: Let the matrix A have identical columns as follows.

Q: Prove that if A is an n X n matrix, then A - Ar is skew-symmetric

A:

Q: Let A be an m × n matrix. Show that A is row equivalent to O if and only if A = O .

A: Consider the given information as A be an n×nmatrix

Q: Find the orthogonal complement to the row space of A and its dimen- sion.

A: Given , A = 125213037 Then , the now-space is span of                   (1,2,5)  ,  (2,1,3)  and…

Q: Prove that if two rows (columns) of the n × n matrix A are proportional, then det(A) = 0.

A: Consider the n×n matrix A whose two rows consider R1 and R2 are proportional.…

Q: Show that if A is both diagonalizable and invertible, so is A-1.

A:

Q: Find a matrix A such that [is: is a solution for AX=X'.

A:

Q: Suppose B = QRQ-1, where Q is orthogonal, and R is upper triangular. Show that if B is symmetric,…

A: Given that B=QRQ-1 where, Q is orthogonal and R is upper triangular. Then, multiply Q on both sides:…

Q: A" A is invertible if and only if A has linearly independent columns.

A: ATA is invertible if and only if A has linearly independent columns.

Q: linear algebra question.

A:

Q: Prove that, for square matrices A and B, AB = BA if and only if (A B)(A + B) = A2 - B2.

A: Given: for square matrices A and B, AB = BA if and only if (A B)(A + B) = A2 - B2.

Q: An n*n matrix C is said to be symmetric if C' = c Select one: True False

A:

Q: Suppose A is an m x mateix whose columns are linearly independent. What is the nullity of A?

A:

Q: Use an augmented matrix and row operations to find A-1, if it exists. (If the inverse does not…

A: find inverse

Q: Use an augmented matrix and row operations to find A-1, if it exists. (If the inverse does not…

A: Our objective is to find inverse of A using augmented matrix and row operations.

Q: If A and B are m × n matrices such that Col(A) and Col(B) are isomorphic, then the RREF of both…

A: If two vector space is Isomorphic then the dimension of the vector space must be same. The dimension…

Q: Find a matrix P such that P

A:

Q: Let A = Then a) A is unitarily diagonalizable and can be factored as

A:

Q: Let A and B be equivalent square matrices. Prove that A is nonsingular if and only if B is…

A: Given, A and B are equivalent square matrices. Let P be any non singular matrix of the same…

Q: Let U be the row-echelon form of square matrix A. prove that if the first r columns of U are…

A:

Q: Prove that, for square matrices A and B, AB = BA if and only if (A - B)(A + B) = A2 - B2

A:

Q: Prove or disprove: If the columns of a square n ×n matrix M are linearly independent, so are the…

A: Linearly independent and dependent vectors

Q: Show that if A is any m × n matrix, then Im A = A and A In = A.

A: Given: Matrix A of order m × n.

Q: / Find a Row A, CoL A, Null (A), whore. A= bazis and dimension Lor 2. 1- -31 -2

A: we have to solve given problem:

Q: Prove that if a row or a column of an n × n matrix A consists entirely of zeros, then det(A) = 0.

A:

Q: Show that if M, - M. - (C ). (6 ). Ms - (: :). M2 = %3D %3D then the span of {M1, M2, M3} is the set…

A:

Q: Suppose A=QR, where Q is m×n and R is n× n. Show that if the columns of A are linearly independent,…

A:

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

A:

Q: Prove that  A + At is symmetric for any square matrix A.

A: Known facts:

Q: Find a basis for R³ including as many columns of A as possible.

A: First we will find the column space of A and then we will extend the column space of A into the…

Q: A matrix Q ϵ Mnxn(R ) is called orthogonal if QQt = I . Prove that if Q is orthogonal, then det(Q) =…

A: Given data: The orthogonal matrix is given by "Q". The given condition is  QQT = I.....(1) Now, take…

Q: Prove that if both products AB and BA are defined, then AB and BA are square matrices.

A:

Q: Suppose that A and B are square matrices with the property AB = BA. Show that ABn = BnA for every…

A:

Q: Let M be a (3x3) matrix such that the dimension of its row space is 1. Which of the following is…

A:

Q: While trying to invert A, [A I] is carried to [P Q] by row operations. Show that P=QA.

A: [A I] is carried to [P Q] by row operations. So, A is carried to P and I is carried to Q. Since A is…

Q: For a square matrix A and a nonnegative integer p, show that (AT )p = (Ap)T .

A: Given: Let A=a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮an1an2⋯annn×nbe a square matrix. Let p be a non-negative…

Q: Use the theorem given below to show that if two rows of a square matrix are equal, then det A = 0.…

A:

Q: ve bases for row(A), col(A), and null(A): -- 1 1 A = 0 1 -1 0 1 -1 -1 w(A):

A:

Q: For matrices M and N, the 3rd column of NM is all zeros. Matrix M has 5 columns and none of them are…

A: Please find the detailed solution in next steps

Q: Show that for any matrices A E RmXn, BE R"XP, ||AB||F < ||A||2||B|F Hint: Write B = [b] ...bp] and…

A: It is given that A∈ℝm×n,B∈ℝn×p Let A=a1Ta2T⋮amT where ai's are m column vectors corresponding to…