Suppose A is invertible and orthogonally diagonalizable. Explain why A–1 is also orthogonally diagonalizable.

Q: Find a basis for the space of 2 × 2 lower triangular matrices. Basis =

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Q: Let A=[124] Recall that row(A) and nul(A) are orthogonal complements. So we can split any x E R²…

A: Since the question has multiple subparts, as per the company guidelines, we will solve the first…

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: Find u and usuch that the null space for the matrix U= 3 v= 0 -2 1 1 -3 -15 1 - 9 1 is span{u, }.

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Q: 5) Prove that the map T : R³ → R³ defined by T((x1, x2, x3)) = (2x1 – 3x2 + 8x3, 4x2 – x3, 2x1) - is…

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Q: Show that any two rows of a matrix can be interchanged by elementary row transformations of the…

A: According to the question,

Q: Prove that the n × n identity matrix is orthogonal.

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Q: If A is diagonalizable then A´is diagonalizable. Select one: O True False

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Q: Algbra prove that A is invertible if AT(a transpose)+A is a positive definite matrix.

A: Given that If A^T+A is a positive definite matrix For a positive definite matrix Det(A^T+A)>0. So…

Q: Show that every square matrix A can be factored as A = RQ, where R is symmetric, positive…

A: Given: Square matrix, AWe need to show that every square matrix, A can be factored as A=RQ where -…

Q: A transformation is defined by the 2 x 2 matrix —а A a + b а where a and b are scalars. If n is an…

A: Solution: The given matrix is A=-ab-aa+ba where a and b are scalars. We have to prove that for any…

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

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Q: Suppose that A and B are invertible 2 x 2 matrices such that det(AT) = 2 and det(B) = -2. Let x ER…

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Q: Show that if A is invertible and DA = CA, then D = C.

A: Given that, A is invertible and DA=CA.

Q: Exercise 51. Is there a square matrix X with real entries such that det(X¹X) = -3?

A: detXTX=-3 X is a square matrix with real entries.

Q: Show that T is a linear transformation by finding a matrix that implements the mapping. Note that x,…

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Q: Let X ∈ R3 be some xed vector with coordinates X1, X2, X3. Consider the action on vectors of the…

A: From the given information we get:

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

A: See below the explanation

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

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Q: Suppose A is invertible.  Explain AT A is invertible.

A: The transpose of an invertible matrix is also invertible, and its inverse is the transpose of the…

Q: Let A be an orthogonal matrix. Show that cA is orthogo- nal if and only if c = ±1.

A: Given:            A be an orthogonal matrix.

Q: Find Jordan form of this matrix

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Q: Let A,B ∈ Matn(F ) denote two square matrices. Assume that A and B are row equivalent. Show that B…

A: The objective of this question is to prove that if two square matrices A and B are row equivalent,…

Q: Let A be a 2 x 2 matrix, and suppose rank(A) = 1. Prove that the rows of A are scalar multiples of…

A: To prove that the rows of matrix A are scalar multiples of each other given that rank(A) = 1, let's…

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: Using the Rank-Nullity Theorem, explain why an n x n matrix A will not be invertible if rank(A) < n.

A: The Rank-Nullity Theorem states that for a linear transformation T:V→W between finite-dimensional…

Q: Let L be the linear operator on R2 that rotates a vector by 30◦ in the counterclockwise direction…

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