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

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: Construct a 3 x 3 triangular matrix A so that 1] the vector 2 is in Col A. 3]

A: We know every lower or upper traingular matrix is a triangular matrix.

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

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

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

A:

Q: Let A = [aj be a 4 x 4 matrix. Develop the general expression for det(A).

A: Given:               Matrix A is : [aij] is the 4×4 matrix;   General expression for the determinant…

Q: and = %3D are oignualus of a matrix A Corraspondeng to the egnualuong=2 and >=-2 then = ACTa t Ja)=…

A: For a given matrix A if 'x' is eigen vector of matrix A corresponding to the eigen value ' q ' then…

Q: Prove the theorem: AAT is a symmetric matrix for any matric A.

A:

Q: Show that if A is an upper triangular matrix, then A T is lower triangular.

A: A is an upper triangular matrix. That is, Now let A=aij be a n×n upper triangular matrix. Then,…

Q: The set of all real two-dimensional row matrices {la b]} with standard matrix addition but scalar…

A: We identify the property that 1.x=x

Q: Let A Mn(C) be a Hermitian matrix. &

A:

Q: If Q is an orthogonal matrix, prove that any matrix obtained by rearranging the rows of Q is also…

A: The matrix is said to be an orthogonal matrix if the product of a matrix and its transpose gives an…

Q: Let A be an invertible matrix and let Az = b (A+AA)(x+ Ar) = b. inequality ||AT|| A

A:

Q: Prove that if A is a nonsingular diagonalizable matrix, then A−1 is also diagonalizable.

A: Suppose A is non-singular then diagonalizable matrix. It implies that there exists an invertible…

Q: Show that eA is nonsingular for any diagonalizable matrix A.

A:

Q: 4. Prove that if A is a square matrix, then det(A™ A) = det(AA") %3D

A:

Q: Let U be an n × n non singular matrix with non zero diagonal entries. Explain why U must be non…

A: Consider the provided question, Let U be an n x n non-singular matrix with non-zero diagonal…

Q: Show that if A is an orthogonal matrix, then A−1 is orthogonal.

A: Explanation: Given that,  A is an orthogonal matrix.

Q: let A nonsingular matrix, c #0, c scalar (cA)" = CA -1 -1 %3D Select one: True O False

A:

Q: Prove that if an upper triangular matrix is orthogonal, then it must be a diagonal matrix

A: Solution: If the multiplication of matrix with its transpose is equal to the identity, then a matrix…

Q: Prove that if A is a nonsingular upper triangular matrix, then A−1 is upper triangular.

A: we have that A is a nonsingular upper triangular matrix and A-1 be its inverse.

Q: The set of all real two-dimensional row matrices {[a b]} with standard matrix addition but scalar…

A: our objective is to conclude

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

A:

Q: Let A = [aij] given by a x bij = (i-j)^3 is a: * Anti-Symmetric matrix Symmetric matrix Identity…

A:

Q: Prove If U is a unit upper triangular matrix, then U is nonsingular and U−1 is also unit upper…

A:

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

A:

Q: Let V = {Set of all invertible N × N matrices} with the standard definition of matrix addition and…

A: A set of vectors V defined on a field F with two operations  vector addition and scalar…

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

A: We have to prove that a square matrix A is invertible if and only if ATA is invertible. Use the…

Q: A symmetric and idempotent matrix is positive semi-definite,

A:

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

A:

Q: Let A be an Hermitian matrix and let B = iA. Show that B is skew Hermitian.

A:

Q: Let A be an n x n Hermitian matrix and X and Y be n x 1 column vectors. Prove that (AX)- Y- X- (AY)…

A: Prove.

Q: b) If the matrix has an A = PDPT, prove that A is a orthogonal diagonalization šymmetric matrix

A: If the matrix has an orthogonal diagonalization A= PDPT, then to show A is a symmetric matrix…

Q: Show that the diagonal entries of a Hermitian matrix must be real.

A:

Q: Prove det(A) = det(A'), for an upper triangular matrix A.

A:

Q: (b) By replacing A with another matrix B different from A in only a single entry, define an inner…

A: To prove a function to be an inner product, we have to show it satisfies the following five…

Q: Let A, B, C be invertible matrices of order 3x3. Show that if AB = AC , then B =C. %3D

A: Let A, B, C be invertible matrices of order 3×3. To show that if AB=AC, then B=C.

Q: If B is invertible matrix the show det B = det B that %3D

A:

Q: Prove that ||A||₁, ||A||, || A||₂, are matrix norms.

A:

Q: 1 Determine whether the matrix C = is an orthogonal matrix. 1

A:

Q: Explain why for any matrix A, AA' exists and prove that AA' is a symmetric matrix

A: Let A is any matrix of order m by n Am×n    and   Atn×m Now  AAt exists of order m by m To prove…

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: Let A be an n x n matrix. Then A is nonsingular if and only if its column vectors form a basis for…

A: We have to check

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: Prove that the symmetric matrix is diagonalizable.

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

Q: Show that if B is a symmetric nonsingular matrix, then B2 is positive definite.

A: Concept: A branch of mathematics which deals with symbols and the rules for manipulating those…

Q: If AC is an invertible matrix, then show that A is also an invertible matrix

A: Given- AC is an invertible matrix, i.e.- ACAC-1=I⇒ACA-1C-1=I⇒AA-1CC-1=I

Q: Prove that A4, A||, ||A||₂ are matrix norms.

A: Given: ∥A∥1, ∥A∥∞, ∥A∥2 To prove ∥A∥1, ∥A∥∞, ∥A∥2 are matrix norms.

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…