Let U be an orthogonal matrix. Prove that det(U) is either 1 or –1.

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

A:

Q: Find the coordinate matrix of x in Rn relative to the standard basis.x = (7, −4, −1, 2)

A:

Q: Show that if A is a symmetric matrix, then AT is symmetric.

A: we have to show that if A is a symmetric matrix, then AT is symmetric.   as we know that a matrix B…

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

A:

Q: Find the coordinate matrix of x relative to the orthonormal basis B in R". . {(금 승, 이 (-승릉이 0.0, 1}…

A:

Q: Let A be an 3 x 4 matrix. What must c and d be if we define the linear transfor- mation T: R → Rd by…

A: Since you have posted multiple questions and we can answer one question. So, answered the first…

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 Amxn matrix , c scalar If cA=0, then Select one: a. c=0 and A=0 b. c=0 or A=0 C. C0 or A=0

A: WE have to check the meaning of cA=0.

Q: Assuming that b # 0, find a matrix that orthogonally diagonalizes a= [: :] P

A:

Q: Let x = (1, 2,−2)T . Find a Householder matrix H such that Hx is a vector of the form (r, 0, 0)T .

A:

Q: Find the matrix C such that BC = AT

A: Let A be an n×n matrix, Then the transpose matrix of A is also an n×n matrix denoted by AT. Now,…

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: Let A be an m x n matrix. Prove that if c is any nonzero scalar, thenrank(cA) = rank(A).

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: Show that for an invertible matrix A: 1 det (A-1) det (A)

A: Given that, the matrix A is invertible. It is known that, detI=1 and detA·A-1=detI.

Q: * Let A be and m x n matrix. Describe the matrix E such that EA is A with row swapped.

A: Given matrix A is a m×n matrix. And the elementary matrix E and EA are row swapped. Since elementary…

Q: Let f(x,y) =x²+y² +cxy be a function on domain R?; c is an arbitrary constant number. (a) Write down…

A:

Q: Find the coordinate matrix of x in Rn relative to the standard basis.x = (5, −2)

A:

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

A:

Q: Show that if A is an orthogonal matrix, then det(A) = ±1.

A: Given If A is an orthogonal matrix. To show det(A) = ±1

Q: Find the coordinate matrix of x in Rº relative to the basis B'. B' = {(3, 0), (0, 8)}, x = (24, 16)…

A: Given the basis B'=3,0,0,8  and x=24,16 Let x1, x2 is coordinate vector that is xB'=x1x2 Then it…

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

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: Compute det(Ei) if Ei is an elementary matrix of type i.

A: Elementary matrix is the type of matrix which is formed by changing one row of the identity matrix.

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

A:

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

A:

Q: Let A and B be orthogonally similar matrices. Prove that AT and BT are orthogonally similar.

A:

Q: Prove that if A is a positive definite matrix and A = UΣV* is a singular value decomposition of A,…

A: A  is a positive definite matrix and

Q: Let f(u) = u™ Au, where u e Rd and A e Rd×d is a symmetric matrix. Calculate Vuf(u).

A: Directional derivative: If the function f is differentiable at x, then the directional derivative…

Q: Show that if A is an orthogonal matrix, then det(A) = +1.

A: To prove, detA=±1

Q: (a) Let M be a square matrix such that M² = 0. Let t, c be scalars and c‡0. Show that

A: We shall solve first question only as you have asked more than one different question - for others…

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

A:

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: Show that if A is an n x n symmetric matrix, then |(Ax)•y = x•(Ay) for all x, y in R".

A: In symmetric matrix, transpose of the matrix is equal to the matrix. AT=A

Q: Let A be a nonsingular matrix. Show that A−1 is also nonsingular and (A−1)−1 = A.

A:

Q: Let A be an n x n matrix. If vi, v2,..., v are vectors in R" such that {Avi, Au2,..., Avn} is…

A:

Q: Prove that A+ A' is a symmetric matrix, w

A:

Q: Let A = (a;;)ij be an m x n matrix of real numbers, and let f : R" Rm be defined by 0-0 x2 x2 f = A

A: To find the derivative of f given: matrix A which has m rows and n columns  also the function f…

Q: Let a be A square matrix of order n show that det(A)=0 if and only if there exist a non zero vector…

A:

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 u is a nonzero vector in R", then uu is a rank one matrix. O True O False

A: We have given u is a non-zero vector in ℝn. Then, u=a1a2..an where, at least one  ai≠0 and 1≤i≤n.…

Q: Let U be an n x n matrix such that UT U=I. Show that U=+-1. (plus or minus 1)

A:

Q: Prove the following statement: If the columns of A are linearly independent, then AT A is…

A: A is invertible if there exists a matrix A-1 such that AA -1=A -1A=I   The vectors v1,.......,vn are…

Q: Show that if A is an orthogonal matrix, then AT is also orthogonal.

A:

Q: Prove that if A is an orthogonal matrix, then ∣A∣ = ±1.

A: Given: A is orthogonal matrix   To Show: A=±1

Q: Let A be an n xn Hermitian matrix and let æ be a vector in C". Show that if c= xAx, then c is real.

A:

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