A matrix Q ∈Mn×n(R) is called orthogonal if QQt= I. Prove that if Q is orthogonal, then det(Q) = ±1.

Q: For A a square matrix such that ATA = I show that det A = ±1.   If A is square and A5 = 0 show…

A: Given, ATA=IdetATA=detIdetATdetA=1  Since, detI=1detA·detA=1    Since, detAT=AdetA2=1detA=±1 This…

Q: The matrix A E M(n, R) is invertible, such that A³ = A. Prove that det(A) = +1.

A: The matrix A∈Mn, ℝ is invertible, such that A3=A. To prove det(A)=±1

Q: Find a matrix A such that e [-] -2 is a solution for Ax= x'.

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Q: 1 0 2 -1 0 2 0 1

A: Given the matrix A=1020-10201 To find an orthogonal matrix P and a diagonal matrix D such that…

Q: Given the partitioned matrix A = [ B]. B]₁ C D in which B and C are invertible. Which of the…

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Q: Let (A.x, Ay, As) solve AT I JI Ax Ay I.l - S 0 X As Assume ri + 0, S and X are positive definite…

A: As per the question we are given a linear system. From which we have to find ∆x , ∆y , ∆s And prove…

Q: For matrix A and a scalar k, (kA)T is equal to: k(A^T) k(A) (k^2) x (A) (k^2) x (A^T)

A: We will solve the matrix operation.

Q: Given the bases B={U , u2} and B'={u'1, u'2} for R2, where and u'1= u'2 = The transition matrix Pa-g…

A: The given bases are B={u1,u2} and B={u1',u2'} where u1=10  ,u1=01   and  u1'=-10  ,u1'=01 . To find…

Q: Given that A, B, C are all square matrices of the same size, matrices A and C are both orthonormal,…

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Q: Given two m ×m matrix X and Y , where XY = Y X. 1) Let u be an eigenvector of X. Show that either Y…

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Q: find the Lineartransfromation T:Maxz (R) /R whose matrix MT with vegard tothe base and the base…

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Q: Under what condition is the function harmonic? Here A = (aij ) is a given, symmetric 3 × 3 matrix.

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Q: Consider the 3 x 3 matrix A and the vector & e R³ given below 0 1 0 1 0 1 0 1 0 1 A = i = Prove that…

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Q: 11. Let A be an m x n matrix and let x, y e R". Prove that if Ax = 0 and y = ATb for some b e R",…

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Q: 2 Consider the 3 x 3 matrix B = 1 2h -1 1 1

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Q: If A is a skew-symmetric n X n matrix and n is odd, prove that det A = 0.

A: Since A is a skew-symmetric matrix we have,  AT=-A where AT is the transpose of matrix A. Then we…

Q: If A is an n × n matrix, show that (Ak )T = (AT )k for any positive integer k.

A: Matrix: We will solve it by the method of induction:(Ak)T=(AT)k case 1: k=1case 2: k=ncase 3: k=n+1

Q: Let A be an nxn matrix. If A-= det A adj A is computed, what should AA be equal to in order to…

A: Using the given matrix equation we need to find the value of AA-1 .

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

A: Given:0111 over ℤ2

Q: Prove that with gauss jordan! If A is an n squared matrix and if we can find a matrix B such that…

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Q: A matrix B in Mnxn (C) is called skew-symmetric if Bt= -B. Prove that if B ϵ Mnxn (C) is…

A: Let B be skew-symmetric matrix of order n, where n is odd, then we have

Q: Verify by finding the matrix of S 0 T (a) by direct substitution and (b) by matrix multiplication of…

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Q: Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). (If an answer…

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Q: Use the invertible matrix theorem to determine the value(s) of A for which the matrix 4. 1A0 320 A-…

A: Given :A=1λ0320121

Q: For a nonsingular matrix A and nonzero scalar k, show that (k A)−1 = 1 k A−1.

A: Given that A is nonsingular matrix. That is, A≠0. Since k is a non zero scalar, kA≠0. Therefore, kA…

Q: If o is a normalised fundamental matrix of X = AX at to = 0 and f is continuous in t, then Xp(t) = |…

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Q: If a is skew symmetric nxn matrix and n is odd,prove that det A = 0.

A: Consider a skew symmetric n x n matrix A where n is odd. Here the objective is to prove that det(A)…

Q: Let A be 2 x2 matrix with tr(A)=4, det(A)=5,then the characteristic polynomial of A is: Select one:…

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Q: Show that for any n x n matrix A there exists another n x n matrix M such that MAM-1 = AT, where AT…

A: Consider the given information. Let the matrix A is an n×n matrix and M is a m×m matrix. And the…

Q: Suppose A is a 2 x 2 matrix such that for all x, y E R², (Ax) · (Ay) = x ·y Must it be true that A…

A: Please comment if you need any clarification. If you find my answer useful please put thumbs up.…

Q: Prove that the following is false by giving a counterexample. а ь с a b If matrix de fis invertible,…

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Q: Consider M as an n x n matrix, we can say that M is nilpotent  if Mx = 0 for some x ∈ +Z . Now:…

A: (a) Let M be a nilpotent matrix .  So, using definition of nilpotent, Mx=0 for some x∈ℤ+ ⇒Mx=0⇒Mx=0…

Q: Find a 3 x 3 matrix A (with real number entries) whose characteristic polynomial is det(A – rls): *…

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Q: Explain why the columns of an nxn matrix A span R" when A is invertible. Choose the correct answer…

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Q: Show that if an n × n matrix A is positive definite, then there exists a positive definite matrix B…

A: Consider the provided question,

Q: For every square matrix A and every scalar c, we have det(cA) = c det(A). O True O False

A: Given statement is  For every square matrix A and every scalar c , we have det ( cA ) = c det ( A )…

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: Suppose V and W are finite-dimensional and T E L(V,W). Show that with respect to each choice of…

A: Suppose V and W are finite-dimensional and T∈LV,W. To show: to each choice of bases of V and W, the…

Q: For a non-singular matrix A, A^(-1) is equal to: (adj(A)) / det(A) det(A) x (adj(A)) det(A) x A

A: Explained below

Q: Let A be a nilpotent matrix (that is, Am = 0 for some m > 1). Prove that if A is diagonalizable,…

A: Consider a nilpotent matrix A. So Am=0 for some m>1.

Q: If A and B are invertible matrices and c is any scalar, then which of the following does not hold?

A: A and B are invertible matrices then  (AB) -1  = B-1 A-1

Q: Show that if the non-zero vectors s1), s(2),..., s) are conjugate to a positive definite matrix G,…

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Q: Show that if A is any n × n matrix, then (a) A + AT is symmetric. (b) A − AT is skew symmetric.

A: Symmetric matrix: Any matrix A is symmetric matrix if and only if it is equal to its transpose that…

Q: Suppose that A is a 4 x 4 matrix such that det A = 10. Enter the value of det(A³).

A:

Q: Suppose matrix A is m×n and B is n×p. Prove that if the columns of B are linearly dependent, then…

A: Let A be m by n matrix and B be a n by p matrix.

Q: et A be 2 x2 matrix with tr(A)=4, det(A)=5,then the characteristic polynomial of . elect one: a. pA)…

A: option a is correct . solution follow as

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

A: Given a matrix, Let A=0111 over Z2 To find the inverse of A.

Q: If A is an invertible matrix and det(A) = 7, what is det (A-1)?

A: Given: If A is an invertible matrix and det(A) = 7 To find: what is det (A-1 )?