Let U and V be n × n orthogonal matrices. Explain why UV is an orthogonal matrix. [That is, explain why UV is invertible and its inverse is (UV)T .]

Q: Find two 3x3 invertible matrices A and B, such that A+B is not invertible.

A:

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: Let E and F be n × n elementary matrices and let C = EF. Is C nonsingular? Explain.

A: Let E and F are elementary matrices and let

Q: Consider the given matrix. Determine det(A) and the inverse matrix of A.

A: Here we have to find the determinant and inverse of a matrix.

Q: Find x such that the matrix A is equal to its own inverse.

A: We have A=-2-33x Now, A-1=1AadjA For a 2×2 matrix A=abcd, we have A=ad-bc  , adjA=d-b-ca Hence, for…

Q: Let A and B be positive definite symmetric n X n matrices and let c be a positive scalar. Show that…

A:

Q: If A is an invertible matrix, explain why A and AT must have the same null space. Is this true if A…

A: If A is invertible matrix then it has trivial null space and a matrix is invertible iff AT is…

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: Let A and B be invertible 5-square matrices such that A| = 4 and B = 2. Then |2A. B 2 %3D

A:

Q: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists)

A: Given matrix is,

Q: decide if the given statement is true or false,and give a brief justification for your answer.If…

A:

Q: Determine whether th e given matrix is orthogonal. If it is, find its inverse

A: Given: To Find: (a). Whether the matrix is orthogonal. (b). If orthogonal find it's inverse.…

Q: Let X Give an example of two 2 × 2 matrices A and B, neither of which is the identy matrix I, such…

A: Ok

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

A:

Q: Let A and B be two square matrices. Assume that both A and B are invertible. Show that BA is…

A: Note that--- If determinant of a matrix is non-zero then the matrix is said to be invertible and if…

Q: 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: Solving the above matrix problem using row operations

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

A:

Q: Let A and B be square matrices of order 2 such that |B| = 4 .What is ( AB) '

A: To see 2nd step

Q: Find the inverse of the matrix A using the technique covered in class. |1 3 3 A = 1 4 3 %3D 1 3 4

A: Augment with a 3 by 3 identity matrix

Q: Let A and B be symmetric n × n matrices and let C = AB. Is C symmetric? Explain.

A:

Q: Determine the inverse matrix of matrix C.

A:

Q: Determine whether the matrix is symmetric, orthogonal, both, or neither.

A: Now transpose of this matrix is, Therefore A is a symmetric matrix.

Q: Prove that the inverse of an orthogonal matrix is orthogonal.

A:

Q: State whether the matrix is diagonalizable or not. 0 0 7 0 O diagonalizable O not diagonalizable

A:

Q: is an orthogonal matrix? Determine whether or not gonal. Justify your response. -1 2 V3 V10 V15 2.

A: INTRODUCTION: If a square matrix Q is an orthogonal matrix, then it must have the property QTQ=QQT=I…

Q: a) Prove that Hermitian, skew-Hermitian, and unitary matrices are normal.

A:

Q: Find the inverse of the matrix A using row operation if A is a non-singular matrix, where, [1 3 -2]…

A: A=13-223001-1

Q: Determine whether the matrix is orthogonally diagonalizable.

A:

Q: If A and B are diagonalizable matrices, then so is the product AB.

A:

Q: If UT = U-1, then matrix U is an orthogonal matrix. %3D O True O False

A:

Q: Verify that det AB = (det A)(det B) for the matrices in Exercises. -[i ]s- [ 3]

A:

Q: Find conditions on w, x, y, and z such that AB = BA for the matrices below.

A: Here we find the condition w,x,y,z such that AB=BA where A=wxyz , B=11-11

Q: Find the inverse of the matrix by Gauss-Jordan Elimination (using an adjoined matrix). (If an answer…

A:

Q: b) Let A, B, and C be nxn matrices and suppose that ABC is invertible. Which of A, B, and C are…

A:

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: If A is an invertible matrix that is orthogonally diagonalizable, show that A-1 is orthogonally…

A: Let the orthogonal decomposition of A be denoted as A = QTDQ where Q is orthogonal and D is…

Q: Let B = and B' = Find the change of base matrix from B' to B and from B to B'

A:

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

A:

Q: Let U and orthogonal matrices. Explain Why UV is an orthogonal matrix. [That is, explain why UvV is…

A: Since we know that all the orthogonal matrices are invertible         ...........(1) And given that…

Q: State whether the matrix is diagonalizable or not. 0 6 0 0 O diagonalizable O not diagonalizable

A:

Q: Let -1 1 A = -1 1 -1 -3 3 Find an upper triangular matrix which is orthogonally similar to A.

A: An upper triangular matrix is a triangular matrix with all elements equal to below the main…

Q: Prove that if A and B are square matrices and AB is invertible, then both A and B are invertible.

A:

Q: Find 2x2 invertible matrices A and B such that A + B not equal to 0 and A + B is not invertible.

A: Wee need to find matrix A and B and do their addition and find invertible

Q: State whether the matrix is diagonalizable or not. [:] 0 7 0 0 O diagonalizable O not diagonalizable

A: Given matrix is 0700 and now subtract λ from the diagonal elements=0-λ700-λThe determinant is…

Q: Let AA be an n×pn×p matrix. Let BB be a generalized inverse of AA and let CC be another generalized…

A:

Q: Let A and B be square matrices of order 2 such thatA=-7 and B=4.What is (ABY"| %3D %3D

A:

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: Let A and B be similar invertible matrices. Show that the matrices A-1 and B-1 are similar.

A: If A and B be similar invertible matrices then to show that the matrices A-1 and B-1 are similar.