3. Prove that a square matrix A is invertible if and only if ATA is invertible.

Q: Suppose A = PRP–1, where P is orthogonal and R is upper triangular. Show that if A is symmetric,…

A: According to the given information suppose, It is required to show that if A is symmetric then R is…

Q: Prove that if matrix A is diagonalizable, then AT is diagonalizable.

A: We have given a matrix A, that is diagonalizable.

Q: Suppose A and B are n×n matrices such that B is invertible and AB is invertible. Prove that A is…

A: Given that the matrices A and B are of order n×n and both A, AB are invertible.

Q: Is this statement true? It is possible for the square matrix A to be invertible and to have its…

A: Note: The determinant of a square matrix A indicates whether A is invertible or not.Now, det(A)=0 if…

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: Show that the matrix is not diagonalizable.

A:

Q: Given a square matrix A, prove that A is invertible if and only if ATA is invertible.

A:

Q: Show that an n x n nonnegative matrix is transitive if and only if £=1 4* is a positive matrix. -k=1

A: Given that, An nonnegative matrix is transitive.…

Q: Use the Invertible Matrix Theorem to decide if A is invertible

A:

Q: 32. Construct a nondiagonal 2 x 2 matrix that is diagonalizable but not invertible.

A: .

Q: (a) A square matrix M satisfies M = -M. Show that I- M is always invertible.

A:

Q: Is it possible for A3 to be an identity matrix without A beinginvertible? Explain.

A: To explain:It is possible for A3 to be an identity matrix without A being invertible.By definition

Q: 2. Prove that a 2 x 2 matrix A with entries in Z26 is invertible if and only if gcd(det(A), 26) = 1.

A:

Q: Give an example of a 2 x 2 matrix A such that im(A) =ker(A). %3D

A: Null space of a linear transformation: T:V(F)→V'(F) be a linear transformation then the null space…

Q: a If A the matrix A is invertible if and only %3D lc d]° if

A: Find the determinant of the matrix A. A=abcdA=a×d−b×c=ad-bc

Q: 7. Matrix A is said to be involutory if A² = I. Prove that a square matrix A is both orthogonal and…

A: In this question we have to prove that a square matrix A is both orthogonal and involuntary.

Q: Prove that if A is an 2021 x 2021 skew-symmetric matrix, then A must be a non-invertible matrix.

A:

Q: Prove that if A is an invertible matrix and B is row equiva- to A, then B is also invertible. ation

A:

Q: Prove that the main diagonal of a skewsymmetric matrix consists entirely of zeros.

A:

Q: Prove that an upper triangular n ×n matrix is invertible if and only if all its diagonal entries are…

A:

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: Suppose that the square matrix A is invertible. Show that the inverse of A is unique.

A: See below the explanation

Q: 5. Prove that if a matrix A is invertible and diagonalizable, then matrix A-1 is also diago-…

A:

Q: Find three values of c for which the matrix C is not invertible.

A:

Q: 32. Construct a nondiagonal 2 × 2 matrix that is diagonalizable but not invertible.

A:

Q: [1 1 Show that the matrix A is not diagonalizable.

A:

Q: Let B be an invertible matrix. Show that A = BTB is positive definite.

A:

Q: Suppose that A is an n x n matrix. Prove that nulity(a)=nullity(AT)

A:

Q: prove that if A is an invertible matrix then AT is also invertible and (AT)-1= (A-1)T

A: Assume a matrix B. Such that B=A-1

Q: Find the canonical form B and an invertible matrix P such that P−1AP = B.

A: In this question. we find the invertible matrix P and B such that P-1AP=B

Q: Prove that an upper or lower triangular n x n matrix is invertible if and only if all its diagonal…

A: Consider A be a n x n upper or lower triangular matrix

Q: The matrix is invertible if and only if k#

A: An invertible matrix is a square matrix that has an inverse. A square matrix is invertible if and…

Q: Let A and B be n x n matrices.  Suppose that AB = On where On is a zero matrix of size n x n, and A…

A:

Q: If A is a square matrix whose rows add up to the zero vector, explain why A cannot be invertible.

A: Given A is square matrix whos rows add up to zero vector.

Q: An n x n matrix A is invertible if ВА In АВ II

A: Inverse Matrix: Let A be an n × n matrix. If there exists an n × n matrix B such that AB = I = BA,…

Q: Prove that if the determinant of a square matrix "A" does not equal to zero, then zero can not be an…

A: Prove that if the determinant of a square matrix "A" does not equal to zero, then zero can not be an…

Q: Is it true that when an n*n matrix whose columns form a basis of R^n, then A is invertible?

A: We shall answer this in next step

Q: Let A be a nondefective n × n matrix with diagonalizing matrix X. Show that the matrix Y = (X−1)T…

A:

Q: Show that if a square matrix A has two equal columns, then A is not invertible.

A: since not a particular question asked ,as per guidelines solution to only first question (i.e Q1) is…

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: Prove that the symmetric matrix is diagonalizable.

A: Concept: A rectangular array of numbers (or other mathematical objects) for those the operations…

Q: If A is a skew-symmetric matrix then show that AAT = AT A and A² is symmetric. [Note AT denotes the…

A: We will use definition

Q: (b) Show that an n x n complex matrix is unitarily diagonalizable if and only if it is normal.

A: By using Schurs theorem the answer is given as follows :

Q: 2. Let A andP be 4 x 4 matrices such that P is invertible, and PAP-

A:

Q: show that if At exists for matrix A > then A is to

A:

Q: Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are…

A: Show that a lower-triangular matrix L is invertible if and only if its main diagonal entries are…

Q: Prove that if L is an invertible lower triangular matrix, then L-l is also lower triangular.

A: handwriting solution

Q: - L

A: Given that A=42-12 Take  A-λI=04-λ2-12-λ=0(4-λ)(2-λ)+2=08-6λ+λ2+2=0λ2-6λ+10=0λ=6±36-402λ=6±2i2λ=3±i