Page 184 - 35Linear Algebra

184 Determinants

7. Show that if M is a 3 × 3 matrix whose third row is a sum of multiples
of the other rows (R 3 = aR 2 + bR 1 ) then det M = 0. Show that the
same is true if one of the columns is a sum of multiples of the others.

8. Calculate the determinant below by factoring the matrix into elemen-
tary matrices times simpler matrices and using the trick

det(M) = det(E −1 EM) = det(E −1 ) det(EM) .

Explicitly show each ERO matrix.

 
2 1 0
det  4 3 1 
2 2 2

a b x y
9. Let M = and N = . Compute the following:
c d z w
(a) det M.
(b) det N.

(c) det(MN).
(d) det M det N.

(e) det(M −1 ) assuming ad − bc 6= 0.
T
(f) det(M )
(g) det(M + N) − (det M + det N). Is the determinant a linear trans-
formation from square matrices to real numbers? Explain.

a b
10. Suppose M = is invertible. Write M as a product of elemen-
c d
tary row matrices times RREF(M).

i
i
i
11. Find the inverses of each of the elementary matrices, E , R (λ), S (λ).
j j
Make sure to show that the elementary matrix times its inverse is ac-
tually the identity.
i
12. Let e denote the matrix with a 1 in the i-th row and j-th column
j
and 0’s everywhere else, and let A be an arbitrary 2 × 2 matrix. Com-
1
pute det(A + tI 2 ). What is the first order term (the t term)? Can you
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