4. Let 1 A = 8 -1 -1 -4 1 -1 Prove that A is not diagonalizable over R, but is diagonaliz- able over C. (b) Find an invertible matrix X € M3x3(C) such that X-'AX is diagonal.

Q: Let C be an nxn matrix that satisfies C2 -3C–2In 0. Show that C is invertible and C=1 = }(31-A)

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Q: Find f(B)

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Q: Let (x, y, z) E R³ and the transformation T: R³ → R² be given by T(x, y, z) = (2x + 4y, x + 3y + z).

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Q: Let -3 4 A-[ 3³ -1], E-[42], F-[^2} = = = -5 Compute the indicated matrix. (If this is not possible,…

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Q: Let A, B, and A+B_ all be square invertible matrices of the same size. Show that (A +B')' = A(A+B)'…

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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: An upper triangular matrix A is one whose entries a^ij are zero if i > j. Prove that the product C =…

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Q: There exists a 7 x 6 matrix A with rank(A) = nullity(A). A) True B) False

A: Suppose that A  be  the  matrix of size 7.Recall  Rank-nullity theorem:  Let  A   be  a  matrix  of…

Q: Prove why A-1=(ATA)-1AT is correct using the invertible matrix theorem

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Q: Prove that the sum of two symmetric positive definite matrices A and B is positive definite (A+B)…

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Q: -9 4 4 1. Let A = -8 3 4 . Find an invertible matrix P and diagonal matrix D so that A = PDP-1.…

A: See the detailed solution below.

Q: TRUE or FALSE: Every invertible n xn matrix is similar to the identity matrix In.

A: To state true or false given statement.

Q: 34. Is it possible for A³ to be an identity matrix without A being invertible? Explain.

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

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Q: The symmetric matrix corresponding to the quadratic form X1 x2 = x} – 2x1x2 + 2x1x3 – 3x + 4x2x3 –…

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Q: An mxn lower triangular matrix is one whos entries above the main diagonal are 0's. When is a sqaure…

A: when the diagonal elements are non-zero then the matrix is invertible. If all the diagonal entries…

Q: 8. Choose one of the following to complete. Extra credit may be earned if you complete both. a)…

A: Given that A and B is invertible. Then AB-1 exists. Also, AB-1=A-1B-1.

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A: First of all to be invertible, a matrix need to be square matrix. So first condition is m=n

Q: Which of the following statements is true for all invertible 3x3 matrices A and B? (3ABA-!)-|B =…

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Q: 5. Show that a complex square matrix is unitary if and only if it is unitarily equivalent to a…

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