Show that a. A is diagonalizable if (a – d)² + 4bc > 0. b. A is not diagonalizable if (a – d)² + 4bc < 0. [Hint: See Exercise 29 of Section 5.1.]

Q: If A is diagonalizable and B is diagonalizable, must it be true that AB is diagonalizable? Give a…

A: If A is diagonalizable and B is diagonalizable then it is not necessary for AB to be diagonalizable.…

Q: Let A be an n×n matrix. Prove that A and AT have the same Jordan canonical form.

A:

Q: 2 If A is diagonalizable then A is diagonalizable. Select one: O True False

A:

Q: In Exercises 9–10, find bases s for the null space and row space of 2. b. А —D | 4 -1 3 -1] 9. а. А…

A: Here in this question we need to find out the basis for null space and row space of the given…

Q: Exercise 2.5.14 While trying to invert A, A I is carried to P Q by row operations. Show that P= QA.

A: To show P=QA if AI is carried to PQ by row operations.

Q: Show that if B is an n × n matrix with entries in C and  B3=B then B is diagonalisable. Where C is…

A:

Q: Let A be an m × n matrix. Prove that N(A) ⊂ N(ATA).

A:

Q: is not diagonalizable (by similarity) but is condiagonalizable. 4.6.P2 Show that (a) [ (b) [] is…

A: To show: (a) The matix i10i is not diagonalizable but it is condiagonalizable. (b) The matix 1-111…

Q: Prove that if A is invertible and diagonalizable, then A-1 is also diagonalizable.

A:

Q: Find two examples to demonstrate that if (a-d)2+4bc =0 then A may or may be diagonalizable

A:

Q: Find the QR decomposition and prove A 3 -5 2.

A: According to our guidelines we can answer only three subparts, or first question and rest can be…

Q: Prove that if A is an invertible n × n matrix, then the rows of A span R“ by using transpose.

A: we use the fact, if A is invertible then A is row equivalent to the identity matrix I.

Q: Show that if A is an n × n matrix, then A = S + K , where S is symmetric and K is skew symmetric.…

A: A matrix A is an n × n matrix, then A = S + K, where S is symmetric and K is skew-symmetric. Also,…

Q: Compute the adjugate of A = the cofactor matrix of A. -2 0 3 5 -2 -2 1 2 noting that adj(A) =…

A:

Q: 2. Prove that a 2 x 2 matrix A with entries in Z26 is invertible if and only if gcd(det(A), 26) = 1.

A:

Q: - Lat A = - 2 - 2 3. 2. 1- - 2 11

A:

Q: Let A be an n x n matrix whose characteristic polynomial splits. Provethat A and At have the same…

A: Let A be an n x n matrix whose characteristic polynomial splits. Prove that A and At have the same…

Q: Prove that k(x,x)=xAx' is a valid kernel, where A is a symmetric positive semidefinite matrix.

A: Let  we consider      kx,x' = xTAx'…

Q: Show that if A is both diagonalizable and invertible, so is A-1.

A:

Q: Suppose A is a complex matrix. Show that AA and A" A are Hermitian.

A: Explanation of the answer is as follows

Q: When A is invertible, MATLAB finds A by factoring LU (where L may be permu - 1 3 -9 6 100 6- 6 A =…

A:

Q: Find a matrix A such that [is: is a solution for AX=X'.

A:

Q: If A is symmetric and invertible and has an LDU factorization, show that U = LT

A: To show: If A is symmetric and invertible and has an LDU factorization, show that U = LT.

Q: , counting the multiplicities. Then A is orthogonally diagonalizable.

A: We have to examine that matrix A is orthogonally diagonalizable or not.

Q: Exercise 6.4.4. Suppose a space X = AUBUC, with A = B = C = CP², AnB= AnC= BnC = S², %3D ANBNC=…

A:

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: Show that if A is skew symmetric, then the elements on the main diagonal of A are all zero.

A: given that A is skew symmetric

Q: 5. Show that if M is a 3 x 3 magic square with weight w, then we can write M as M = Mo + kJ %3D…

A: Given that ,  A       magic square  To show that : We can write as                     Where,  is a…

Q: 1. Is given a nonsingular matrix P. If A is positive semidefinite, then P'AP is positive…

A:

Q: Let A = Find necessary and sufficient condi- d tions for A to be diagonalizable.

A: Given:         A = abcd   To find:      The necessary and sufficient conditions for A to be…

Q: [2 Is the collection of matrices a linearly independent set in M2x2? If so, does the collection…

A:

Q: Show that if a square matrix A satisfies A2 − 4A + I = 0, then A is invertible and A−1 = 4I − A.…

A:

Q: Let A be an n x n matrix such that Am = 0 for some positive integer m. Show that A is not…

A: A is a n×n invertible matrix, such that Am=0⋯⋯1 Taking determinants both side, detAm=det0detAm=0…

Q: If A is invertible and diagonalizable, then A also is diagonalizable. Select one: O True O False

A: Solve for true or false

Q: If A is an n×nmatrix whose LU decomposition is PA = LU, prove that det(A)=(-1)' u where r is the…

A: Given: A is an n x n matrix. To prove: det(A)=-1n×u11 . . . unn

Q: Prove that, if A is similar to B, and B is diagonalizable, then A is diagonalizable.

A: Given that A is similar to B and B is diagonalizable.  To find A is diagonalizable.

Q: Associated with every square matrix A is a real number denoted by |A| called the __________ of A .

A: |A| denotes the determinant of a matrix A. Hence,

Q: §7.2, Exercise 8. Let A be an n × n matrix. Suppose that A² + A+ In = 0. Prove that A is invertible.

A:

Q: Show that if A is both diagonalizable and invertible, then so What does it mean if A is…

A: Introduction: If matrix A is an invertible matrix then every eigenvalue of this matrix non-zero.

Q: Write down a n × n matrix A such that Cond(A) =1, but determinant(A) → 0 as n → ∞.

A: claim- write down a n×n matrix A such that CondA=1, but detA→0 as n→∞. it is known that condition…

Q: 5. Recall that a square matrix is called idempotent if A? = A. Prove that In is the only nonsingular…

A: To prove that In is the only non singular idempotent  matrix

Q: . Prove that Sp(n, R) is an embedded submanifold of Mn(R). What is its dimension? What is its Lie…

A: It is given that

Q: A square matrix is said to be idempotent if A2= A. Prove that if A is idempotent then 2A-I is…

A: If the square of the matrix is equal to the identity matrix then the matrix is said to be its own…

Q: Find the Jordan canonical form J for -10 -2 2 B = -4 2 -1 -4 2 4 -1 1 3 Also, find a matrix Psuch…

A: Given :  B = -10-2-421-x24-42-1-x-42-113-x ⇒p(x)=det(A-xI)=-1-x0-2-4x1-x24-42-1-x-42-113-x p(x)=…

Q: If A and B are both invertible, then A + B is invertible.

A: Given that A and B are invertible matrices.   Invertible matrices:   If A is invertible matrix, then…

Q: Is this statement true or false? EXPLAIN answer with examples, your Every linear transfor mation T:…

A:

Q: Suppose that a matrix M is nilpotent if exists an x element of Z+ in where Mx = 0. If M is…

A: We will prove the given result.

Q: Show that the matrix AT A is exactly singular in floating point arithmetic if A = € 0 where e is a…

A:

Q: If A EM,C {c} such that A2 = A. Prove that A is not a strictly diagonally matr. the last word is…

A: A square matrix is said to be strictly diagonally Matrix if all entries are zero except diagonal…