Page 21 - A Course in Linear Algebra with Applications

1.1: Matrices 5

is lower triangular if all entries above the principal diagonal
are zero. For example, the matrices

are upper triangular and lower triangular respectively.

(vii) A square matrix in which all the non-zero elements lie
on the principal diagonal is called a diagonal matrix. A scalar
matrix is a diagonal matrix in which the elements on the prin-
cipal diagonal are all equal. For example, the matrices

a 0 0 \ fa 0 0
0 6 0 and 0 a 0
0 0 c / \ 0 0 a

are respectively diagonal and scalar. Diagonal matrices have
much simpler algebraic properties than general square matri-
ces.

Exercises 1.1
l
1. Write out in extended form the matrix ((—\) ~^(i + j))2,4-
2. Find a formula for the (i,j) entry of each of the following
matrices:

/ I 2 3 4
- 1 1 - 1 5 6 7 8
(a) 1 - 1 , (b) 9 10 11 12
1
1 - 1 1
- 1 1 - 1
\13 14 15 16