Solved: Write out the details of the proof of Theorem 2.1.3

Let A = 324 1 2 3 232 (a) Find the values of det(M21), det(M22), and det(M23). (b) Find the values of A21, A22, and A23. (c) Use your answers from part (b) to compute det(A).

Use determinants to determine whether the following 2 2 matrices are nonsingular: (a) 3 5 2 4 (b) 3 6 2 4 (c) 3 6 2 4

Evaluate the following determinants: (a)     3 5 2 3     (b)     5 2 8 4     (c)       312 245 245       (d)       430 312 5 1 4 (e)       132 4 1 2 213       (f)       2 1 2 132 516       (g)        2001 0100 1620 1 1 2 3        (h)        2121 3011 1 2 2 1 3231

Evaluate the following determinants by inspection: (a)     3 5 2 4     (b)       200 410 7 3 2       (c)       300 211 122       (d)        4021 5042 2034 1023

Evaluate the following determinant. Write your answer as a polynomial in x:       a xbc 1 x 0 0 1 x

Find all values of for which the following determinant will equal 0:     2 4 3 3

Let A be a 3 3 matrix with a11 = 0 and a21 = 0. Show that A is row equivalent to I if and only if a12a21a33 + a12a31a23 + a13a21a32 a13a31a22 = 0

Write out the details of the proof of Theorem 2.1.3.

Prove that if a row or a column of an n n matrix A consists entirely of zeros, then det(A) = 0.

Use mathematical induction to prove that if A is an (n + 1) (n + 1) matrix with two identical rows, then det(A) = 0.

Let A and B be 2 2 matrices. (a) Does det(A + B) = det(A) + det(B)? (b) Does det(AB) = det(A) det(B)? (c) Does det(AB) = det(BA)? Justify your answers.

Let A and B be 2 2 matrices and let C = a11 a12 b21 b22 , D = b11 b12 a21 a22 , E = 0 0 (a) Show that det(A + B) = det(A) + det(B) + det(C) + det(D). (b) Show that if B = EA, then det(A + B) = det(A) + det(B).

Let A be a symmetric tridiagonal matrix (i.e., A is symmetric and aij = 0 whenever |i j| > 1). Let B be the matrix formed from A by deleting the first two rows and columns. Show that det(A) = a11 det(M11) a2 12 det(B)