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Matrix Algebra problems.

*NOTE*: PLEASE DO NOT POST CODE ANSWERS.

Please show full work, and you must post answers to all the problems. Do not post answer to only 1 problem. Thank you. Will rate kindly.

1. Find the determinants, eigenvalues and eigenvectors of all the matrices below. Check if the determinant equals the product of its eigenvalues and if its trace equals the sum of its eigenvalues. Asl0 2 01, B-10 2 11, c=10 2 11, D-14 3 1 2 -1 2. The following are characteristic polynomials of some matrices. Construct a matrix that corresponds to each characteristic polynomial. What are the eigenvalues of those matrices? 2_4 3. Determinant of matrix A which is of size 10-by-10 is 5. What is the determinant of this matrix after each of the following transformations: - First, its first 2 rows are multiplied by 4 Second, its first column is added to columns 3 and 4 Third, its 3d row is switched with its 6th row Fourth, its 8th row is replaced with its 1st row 4. If matrix A is orthogonal (dot product of each pair of its columns is zero; also dot product of each pair of its rows is 0; dot product of each row or column with itself is 1), show that A- A1. Hint: calculate A'A and AA 5. Show that for a square matrix A of size n, det(cA) c"det(A), where c is any scalar. Hint: use property that multiplying one row of a matrix by c also multiplies the determinant of the matrix by c. 1. Find the determinants, eigenvalues and eigenvectors of all the matrices below. Check if the determinant equals the product of its eigenvalues and if its trace equals the sum of its eigenvalues. Asl0 2 01, B-10 2 11, c=10 2 11, D-14 3 1 2 -1 2. The following are characteristic polynomials of some matrices. Construct a matrix that corresponds to each characteristic polynomial. What are the eigenvalues of those matrices? 2_4 3. Determinant of matrix A which is of size 10-by-10 is 5. What is the determinant of this matrix after each of the following transformations: - First, its first 2 rows are multiplied by 4 Second, its first column is added to columns 3 and 4 Third, its 3d row is switched with its 6th row Fourth, its 8th row is replaced with its 1st row 4. If matrix A is orthogonal (dot product of each pair of its columns is zero; also dot product of each pair of its rows is 0; dot product of each row or column with itself is 1), show that A- A1. Hint: calculate A'A and AA 5. Show that for a square matrix A of size n, det(cA) c"det(A), where c is any scalar. Hint: use property that multiplying one row of a matrix by c also multiplies the determinant of the matrix by c