Linear Programming Lecture Notes - Penn State Personal Web Server

Definition 3.6 (Matrix Transpose). If A ∈ R m×n is a m × n matrix, then the transpose
of A dented A T is an m × n matrix defined as:
(3.5) A T ij = Aji
Example 3.7.
T 1 2
(3.6)
=
3 4
1 3
2 4
The matrix transpose is a particularly useful operation and makes it easy to transform
column vectors into row vectors, which enables multiplication. For example, suppose x is
an n × 1 column vector (i.e., x is a vector in R n ) and suppose y is an n × 1 column vector.
Then:
(3.7) x · y = x T y
Exercise 18. Let A, B ∈ R m×n . Use the definitions of matrix addition and transpose
to prove that:
(3.8) (A + B) T = A T + B T
[Hint: If C = A + B, then Cij = Aij + Bij, the element in the (i, j) position of matrix C.
This element moves to the (j, i) position in the transpose. The (j, i) position of A T + B T is
A T ji + B T ji, but A T ji = Aij. Reason from this point.]
Exercise 19. Let A, B ∈ R m×n . Prove by example that AB = BA; that is, matrix
multiplication is not commutative. [Hint: Almost any pair of matrices you pick (that can be
multiplied) will not commute.]
Exercise 20. Let A ∈ R m×n and let, B ∈ R n×p . Use the definitions of matrix multiplication
and transpose to prove that:
(3.9) (AB) T = B T A T
[Hint: Use similar reasoning to the hint in Exercise 18. But this time, note that Cij = Ai··B·j,
which moves to the (j, i) position. Now figure out what is in the (j, i) position of B T A T .]
Let A and B be two matrices with the same number of rows (so A ∈ Rm×n and B ∈
Rm×p ). Then the augmented matrix [A|B] is:
⎡
⎤
a11 a12 . . . a1n b11 b12 . . . b1p
⎢ a21 a22 . . . a2n b21 b22 . . . b2p ⎥
(3.10) ⎢
⎣
.
. ..
.
⎥
. . .. .
⎦
am1 am2 . . . amn bm1 bm2 . . . bmp
Thus, [A|B] is a matrix in R m×(n+p) .
Example 3.8. Consider the following matrices:
1 2
7
A = , b =
3 4
8
Then [A|B] is:
[A|B] =
1 2 7
3 4 8
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