5 points) Prove that if A is a square invertible n xn matrix satisfying A2 - 3A+ I, = 0, nen A- = 3In - A, where 0 represents the n x n zero matrix and In the n x n identity natrix. Justify/explain your solution.

Q: Let A be an 8 × 5 matrix of rank 3, and let b be anonzero vector in N(AT ). How many least squares…

A:

Q: 8) Show that the matrix -1 Bz 1-4 4 -3 -2 is idempotent.

A: Given…

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: Prove Any n by n triangular matrix with main diagonal entriesare all 1 is diagonalizable if and only…

A: Triangular Matrix: A triangular matrix is a square matrix where all the values above or below the…

Q: 3. Let A be an n x n singular matrix. Describe one way to construct a non-zero matrix B such that AB…

A:

Q: 2) does the following matrix is orthonormal and orthogonal? Why? R = 2

A: Orthogonal and orthonormal matrix

Q: If A * I = A, and A is a 3x5 matrix, what are the dimensions of I? Show complete solution as proof.

A:

Q: Show that matrix A 5 1 0 5 cannot be diagonalized by using (λI-A)

A:

Q: #16

A: Recall the following facts.

Q: [1] Construct a 3 x 3 nonzero matrix A such that the vector |2 is a solution of 3. Ах 3D0

A: A vector is given let x=123 To find: A non-zero 3×3× matrix,A=a11a12a13a21a22a23a31a32a33

Q: 2. Consider the 6 x 8 matrix [1 3 0 0 0 0 1 0 2 0 5 -3 0 0 0 A 0 0 1 4 -7 -9 00 0 0 0 0 0 0 0 0 0 0…

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Q: Find an example of a singular 2 × 2 matrix satisfying A2 = A.

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Q: 4. Prove that the product of two upper triangular matrices is always another upper triangular…

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Q: 5. Prove that if a matrix A is invertible and diagonalizable, then matrix A-1 is also diago-…

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Q: a() – 4 (6) - (1) - (-) - Find a 2 x 2 matrix A such that A

A: We have to find a 2 × 2 matrix A such that A12=A23=41 Let the matrix A = abcd

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: 32. Construct a nondiagonal 2 × 2 matrix that is diagonalizable but not invertible.

A:

Q: Let A be an m x n matrix with rank m and B be ann x p matrix with rank n. Determine the rank of AB.…

A: to find the rank of AB.

Q: Find the rank and nullity of the matrix; then verify that the values obtained satisfy Formula 4 of…

A: Hey, since there are multiple questions posted, we will answer the first question. If you want any…

Q: If A and B are nXn diagonalizable matrices, then A+B is also diagonalizable. True or False? Give an…

A: Given statement, If A and B are n×n diagonalizable matrices, then A+B is also diagonalizable.

Q: Find a (2x2) singular matrix A that maps (3, 1) into (2, 4).

A:

Q: [3 Compute the mnatrix exponential et for the diagonal matrix A = %3D

A: We have to find the exponential matrix of given diagonal matrix

Q: 5. Prove that the cigenvalues of a Hermitian matrix are always real-valued, and that the inner…

A: Both the proofs of the given question has been proved in step 2 and step 3.

Q: Find a matrix P such that P'AP orthogonally diagonalizes A. Verify that P"AP gives the proper…

A:

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: 31. Let A be a 5 ×7 matrix with rank 4. a. What is the dimension of the solution space of Ax = 0? b.…

A:

Q: 2. Construct a 4x4 nonzero matrix A such that the vector -3 is a solution of Ax= 0. -4 2.

A:

Q: Suppose U is mxn matrix. Explain why if U has orthonormal columns, then we must have m greater than…

A: Let U be a m×n matrix.We need to show that if U has orthonormal columns, then we must have m greater…

Q: Let A be 5x1 matrix. Knowing that Rank (A) = 2. Then Nullity(A) is equal to: O 2 ONone of these

A: The solution is given below in the next step:

Q: Let A be a real matrix such that the linear system Ax = 0 has a nonzero complex solution. Show that…

A:

Q: Find the general parametric solution for the augmented matrix form. Identify the homogenous solution…

A: Solution: AX=b A=22341-3,     X=xyz,     b=03

Q: §3.8, Exercise 4. Let A be a 2 × 2 matrix having integer entries. Find a condition on the entries of…

A: According to the given information, it is required to find the condition on the entries of A such…

Q: A = (aij) be a 10 x 10 positive definite matrix with Cholesky factor R = pose that a11 = 9, a12 =…

A:

Q: 3 Let be 4mxn & Bnxm Matrixes that meet: AB = 1 m Prove that to the homogeneous system Bx = 0 There…

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Q: 11. Construct a matrix with the required property or explain why this is impossible. (a) (.-) Row…

A: See the detailed solution below.

Q: Find the singular values the matrix:

A: A singular value of a real matrix A is the positive square root of an eigen value of the symmetric…

Q: .. Let A = [aij] be a 3 x 3 square matrix such that aij = 1 – |i – j|- (a) Construct the matrix A…

A:

Q: Find a (2x2) singular matrix A that maps (2, 2) into (1, 2).

A: Here in this question find a 2×2 matrix that maps (2,2)  into (1,2). Here i attached solution of…

Q: 5. (Optional) Prove that: A symmetric and idempotent matrix is positive semi-definite,

A:

Q: 7. Show that the matrix A=I– 2vv' , where v is a unit vector in R", and I is an nxn identity matrix,…

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Q: Solve the Matrix Equation XAB-BAX=X with A,B,X are nxn matrix with A,X are invetribles.

A: The given matrix equation is, XAB-BAX=X where A, B and X are n×n matrices with A and X invertible.

Q: A square matrix M satisfies MT = -M. Show that I- M is always invertible.

A: Hi there!. You have posted multiple questions in a single request. We will only solve the 1st one.…

Q: §3.6, Exercise 1. Let A be an m × n matrix. Show that the matrix AAt is symmetric.

A: For a matrix X to be symmetric, Xt=X.   Here, A is a m×n matrix, suppose At is its transpose. Claim:…

Q: 2 2 1 2 -1 2 is orthogonal. Verify whether the matrix A = 3. -1 2 2

A: Given matrix is, A=132212-12-122

Q: 6. Suppose that A is an n x n matrix. Prove that nullity(A) = nullity(A™). %3D

A:

Q: Prove that deleting the last column of an echelon form (respectively, the row canonical form) of an…

A: Echelon matrices come in two forms: the row echelon form (ref) and the reduced row echelon form…

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: §8.2, Exercise 4. Let A be a nonzero 3 × 3 matrix such that A? = 0. Show that rank(A) = 1. %3D

A: Given A is a non-zero 3×3 matrix such that  A2=0 To prove the rank of A is 1 For the proof, the…

Q: Determine all n x n matrices that are similar to I. Let A be annx n matrix similar to I, Then there…

A: Given, A be n×n matrix similar to In. Hence, ∃P n×n invertible matrix, such that A = P-1InP