linear-guest

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 elementary
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 ≠ 0.
(f) det(M T )
(g) det(M + N) − (det M + det N). Is the determinant a linear transformation
from square matrices to real numbers? Explain.
( ) a b
10. Suppose M = is invertible. Write M as a product of elementary
row matrices times
c d
RREF(M).
11. Find the inverses of each of the elementary matrices, E i j, R i (λ), S i j(λ).
Make sure to show that the elementary matrix times its inverse is actually
the identity.
12. Let e i j denote the matrix with a 1 in the i-th row and j-th column
and 0’s everywhere else, and let A be an arbitrary 2 × 2 matrix. Compute
det(A + tI 2 ). What is the first order term (the t 1 term)? Can you
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