5. Let A be an n x n matrix. (a) (b) Show that if A is an orthogonal matrix, then det(A) = ±1. Show that if A 0 and A³ = 0, then A is not diagonalizable. (Hint: Assume A is diagonalizable and then get a contraction.)

Q: Let A be an invertible matrix. Show that det (adj (A)) = det (A-¹).

A:

Q: Prove that A2 # 0. Let A € M4×4(F) be a diagonalizable matrix with rank 1.

A:

Q: Let A be any invertible 9 x 9 matrix. Which of the following is equal to det(Adj A)? OA. (det A) B.…

A: Given: A is an invertible matrix of order 9×9 We are required to find: det(AdjA)

Q: Let A be a matrix whose entries are integers, and det A=1. 4 : Show that the entries of A-1 are all…

A: given ,det A = 1

Q: Let A be a 3 x 3 symmetric matrix given by A = 2 -1 0 -1 0 2 -1 2 -1 Find an upper triangular matrix…

A:

Q: Q9. As usual, let In denote the n x n identity matrix. 1 - (a) Find a 2 × 2 matrix such that A² =…

A:

Q: Let A be a 3 x 3 matrix such that det(A) = 2. Evaluate det(Aª) and det(2A).

A: Recall the following. If A is a n×n matrix and m be any positive scalardet k A=kndet Adet Am=det Am

Q: Determine whether A is diagonalizable. A = (D, P) = 0 2 2 220 OYes O No Find an invertible matrix P…

A:

Q: Suppose A is a symmetric matrix. (Au)" : u"A

A:

Q: 3) Let A = 2 -1 Find a matrix P that orthogonally diagonalizes A. Use P to com- 0 -1 pute A5.

A: Given, A=11012-10-11first we have to find a matrix Pfor eigen values of…

Q: Exercise 2. Suppose that A and B are matrices of the same size. Prove the following: (a) (aA)¹ = aAª…

A:

Q: Let A be a square matrix such that A = 0. Show that 4 : (I – A)-1 = I+ A+ A? + A³. Here I is the…

A:

Q: diag(a1, ..., an) be a diagonal matrix (i.e., the matrix with the numbers a1, ..., an on the main…

A:

Q: Let A be an n by n matrix. (a) Show that Nul(A - I) C Col(A). (b) If A² = A holds. Prove that Col(A)…

A: To show that Nul(A - I) is a subset of Col(A), we need to prove that every vector in Nul(A - I) is…

Q: Let A be an n × n matrix with a column of zeros. Show directly from the Leibniz formula that…

A: Let, A be an n×n matrix with a column of zeros.  To prove detA=0 using Leibnitz formula for…

Q: Prove: For any square matrix A, det(A) = det(A¹).

A: We know Transpose of matrix is found by interchanging its rows into columns or columns into rows

Q: Let A be an invertible, n x n matrix such that A² = A. a) Calculate det(A). b) If n = 3, what is A?…

A: