Page 213 - Advanced engineering mathematics

7.1 Matrices 193

This is routine to prove. Note that the dimensions must be correct - we must multiply A on
the left by I n , but on the right by I m , for these products to be defined.

EXAMPLE 7.6
Let
⎛ ⎞
1 0
⎝ 2
A = 1 ⎠ .
−1 8
Then
⎛ ⎞⎛ ⎞ ⎛ ⎞
1 0 0 1 0 1 0
I 3 A = 0 1 0 ⎠⎝ 2 1 ⎠ = ⎝ 2 1 ⎠ = A
⎝
0 0 1 −18 −1 8
and
⎛ ⎞ ⎛ ⎞
1 0 1 0
10
AI 2 = ⎝ 2 1 ⎠ = ⎝ 2 1 ⎠ = A.
01
−18 −1 8
If A =[a ij ] is an n × m matrix, the transpose of A is the m × n matrix defined by
t
A =[a ji ].
We form the transpose by interchanging the rows and columns of A.

EXAMPLE 7.7
Let

−1 6 3 −4
A = ,
0 π 12 −5
t
a2 × 4 matrix. Then A is the 4 × 2matrix
⎛ ⎞
−1 0
6 π ⎟
⎜
t
A = ⎜ ⎟ .
⎝ 3
12 ⎠
−4 −5
THEOREM 7.3 Properties of the Transpose
t
1. (I n ) = I n .
2. For any matrix A,
t t
(A ) = A.
3. If AB is defined, then
(AB) = B A .
t
t
t
Proof of Conclusion (2) It is obvious if we take the transpose of a transpose, then we inter-
change the rows and columns, then interchange them again, leaving every element in its original
position.
It is less obvious that, if we take the transpose of a product, then we obtain the product of
the transposes, in the reverse order, which is conclusion (3). We will prove this.

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October 14, 2010 14:23 THM/NEIL Page-193 27410_07_ch07_p187-246