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Special Forms of Matrices

feasible, and will result in a 1 × 1 , that is,

1
⋅
CD = 234 – 1 = 2 () () + 3 () ( ⋅ – 1 + 4 () () = 7
⋅
⋅
)
2
1
2
The dimensions for and are respectively 3 × 1 1 × 3 and therefore, the product DC⋅ is also
C
D
feasible. Multiplication of these will produce a 3 × 3 matrix as follows.
1 1 () () 1 () () 1 () () 2 3 4
⋅
⋅
⋅
2
4
3
⋅
DC = – 1 234 = – ( 1 ⋅ 2 () ( – 1 ⋅ 3 () ( – 1 ⋅ 4 () = – 2 – 3 – 4
)
)
)
2 2 () () 2 () () 2 () () 4 6 8
⋅
⋅
⋅
3
2
4
Check with MATLAB:
C=[2 3 4]; D=[1; −1; 2]; % Define matrices C and D
C*D % Multiply C by D
ans =
7
D*C % Multiply D by C
ans =
2 3 4
-2 -3 -4
4 6 8
Division of one matrix by another, is not defined. However, an equivalent operation exists, and it
will become apparent later in this chapter, when we discuss the inverse of a matrix.

4.3 Special Forms of Matrices
† A square matrix is said to be upper triangular when all the elements below the diagonal are
zero. The matrix below is an upper triangular matrix.
A

a 11 a 12 a 13 … a 1n
0a 22 a 23 … a 2n
A = 0 0 …… … (4.4)

…… 0 ……
0 0 0 … a mn

Numerical Analysis Using MATLAB® and Excel®, Third Edition 4−5
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