25 pl ces.tudyfore5.3 Diagonalization 28924. A is a 3 x 3 matrix with two eigenvalues. Each eigenspace isone-dimensional. Is A diagonalizable? Why?25. A is a 4 x 4 matrix with three eigenvalues. One eigenspaceis one-dimensional, and one of the other eigenspaces is two-dimensional. Is it possible that A is not diagonalizable?Justify your answer.26. A is a 7 x 7 matrix with three eigenvalues. One eigenspace istwo-dimensional, and one of the other eigenspaces is three-dimensional. Is it possible that A is not diagonalizable?Justify your answer.27. Show that if A is both diagonalizable and invertible, then sois A-¹.28. Show that if A has n linearly independent eigenvectors, thenso does AT. [Hint: Use the Diagonalization Theorem.]29. A factorization A = PDP is not unique. Demonstrate this3for the matrix A in Example 2. With D₁ =[³009]5the information in Example 2 to find a matrix P₁ such thatA = P₁D₁P₁¯¯¹.use-130. With A and D as in Example 2, find an invertible P₂ unequalto the P in Example 2, such that A = P₂DP₂¹.74231. Construct a nonzero 2 x 2 matrix that is invertible but notdiagonalizable.32. Construct a nondiagonal 2 × 2 matrix that is diagonalizablebut not invertible.[M] Diagonalize the matrices in Exercises 33-36. Use your ma-'s eigenvalue command to find the eigenvalues, andas in Section 5.1.

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A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other…

25. A is a 4 x 4 matrix with three eigenvalues. One eigenspace is one-dimensional, and one of the other eigenspaces is two- dimensional. Is it possible that A is not diagonalizable? Justify your answer.