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produces

detN =
0
invN
Warning: Matrix is close to singular or badly
scaled.

Homework Problems
Pb. 8.1 As earlier defined, a square matrix in which all elements above
(below) the diagonal are zeros is called a lower (upper) triangular matrix.
Show that the determinant of a triangular n ⊗ n matrix is

det(T) = T T T … T nn
11 22 33
Pb. 8.2 If M is an n ⊗ n matrix and k is a constant, show that:
det(kM) = k det(M)
n
Pb. 8.3 Assuming the following result, which will be proven to you in linear
algebra courses:

det(MN) = det(M) × det(N)

Prove that if the inverse of the matrix M exists, then:

1
det(M −1 ) =
det(M )

8.6 Solving a System of Linear Equations

Let us assume that we have a system of n linear equations in n unknowns that
we want to solve:

Mx + M x + M x +…+ M x = b
11 1 12 2 13 3 1 n n 1
Mx + M x + M x +…+ M x = b
21 1 22 2 23 3 2 n n 2
(8.8)
M
M x + Mx + Mx +…+ M x = b
n1 1 n2 2 n3 3 nn n n

© 2001 by CRC Press LLC