Summary Chapter 2: Vectors and Matrix

Sum of Matrices
Let A=(aij) y B=(bij) d be two m X n matrices. Then the sum of A and B is the matrix
m X n, A + B given by:
𝒂𝟏𝟏 + 𝒃𝟏𝟏 𝒂𝟏𝟐 + 𝒃𝟏𝟐 ⋯ 𝒂𝟏𝒏 + 𝒃𝟏𝒏
𝒂 + 𝒃𝟐𝟏 𝒂𝟐𝟐 + 𝒃𝟐𝟐 ⋯ 𝒂𝟐𝒏 + 𝒃𝟐𝒏
𝑨 + 𝑩 = (𝒂𝒊𝒋 + 𝒃𝒊𝒋 ) = ( 𝟐𝟏 )
⋮ ⋮ ⋮ ⋮
𝒂𝒏𝟏 + 𝒃𝒏𝟏 𝒂𝒏𝟐 + 𝒃𝒏𝟐 ⋮ 𝒂𝒎𝒏 + 𝒃𝒎𝒏


That is, A + B is the m X n matrix obtained by adding the corresponding components
of A and B
The sum of two matrices is defined only when the matrices are of the same size. So,
for example, it is not possible to add the matrices
−1 0
1 2 3
( )y ( 2 −5)
4 5 6
4 7
1
1
or the matrices (vectors)( ) y (2), that is, they are incompatible under the sum
2
3
Example of adding two vectors
2 4 −6 7 0 1 6 −2 2+0 4+1 − 6+6 7−2
( 1 3 2 1) + ( 2 3 4 3 ) =( 1+2 3+3 2+4 1+3)
−4 3 −5 5 −2 1 4 4 −4 − 2 3+1 −5 + 4 5+4
𝟐 𝟓 𝟎 𝟓
(𝟑 𝟔 𝟔 𝟒)
−𝟔 𝟒 −𝟏 𝟗


Multiplication of a matrix by a scalar
If A = (aij) is an m X n matrix and if α is a scalar, then the m X n matrix, αA, is given
by
𝜶𝒂𝟏𝟏 𝜶𝒂𝟏𝟐 ⋯ 𝜶𝒂𝟏𝒏
𝜶𝒂 𝜶𝒂𝟐𝟐 ⋯ 𝜶𝒂𝟐𝒏
𝜶𝑨 = (𝜶𝒂𝒊𝒋 ) = ( 𝟐𝟏 )
⋮ ⋮ ⋮ ⋮
𝜶𝒂𝒏𝟏 𝜶𝒂𝒏𝟐 ⋮ 𝜶𝒂𝒎𝒏
This is αA = (αaij) is the matrix obtained by multiplying each component of A by α. If
Αa = B = (bij), then bij = αaij for i = 1, 2, …., m y j = 1, 2, …., n

, Example
1 −3 4 2
A =( 3 1 4 6 ), Then calculate 2A, (-1/3) A y 0A
−2 3 5 7
1) 2A
1 −3 4 2 2 ∗ 1 2 ∗ −3 2∗4 2∗2
2∗( 3 1 4 6 ) = (2 ∗ 3 2 ∗ 1 2∗4 2 ∗ 6) =
−2 3 5 7 2 ∗ −2 2 ∗ 3 2∗5 2∗7
𝟐 −𝟔 𝟖 𝟒
( 𝟔 𝟐 𝟖 𝟏𝟐 ),
−𝟒 𝟔 𝟏𝟎 𝟏𝟒


2) (-1/3) A
1 1 1 1
(− ) ∗ 1 (− ) ∗ −3 (− ) ∗ 4 (− ) ∗ 2
3 3 3 3
1 1 −3 4 2 1 1 1 1
(− ) ∗ ( 3 1 4 6 ) = (− ) ∗ 3 (− ) ∗ 1 (− ) ∗ 4 (− ) ∗ 6
3 3 3 3 3
−2 3 5 7
1 1 1 1
(− ) ∗ −2 (− ) ∗ 3 (− ) ∗ 5 (− ) ∗ 7 )
( 3 3 3 3
𝟏 𝟒 𝟐
−𝟑 𝟏 −𝟑 −𝟑
𝟏 𝟒
−𝟏 −𝟑 −𝟑 −𝟐 ,
𝟐 𝟓 𝟕
−𝟑 −𝟏 −𝟑 −𝟑
( )


3) 0A
1 −3 4 2 0 ∗ 1 0 ∗ −3 0∗4 0∗2
0∗( 3 1 4 6 ) = (0 ∗ 3 0 ∗ 1 0∗4 0 ∗ 6) =
−2 3 5 7 0 ∗ −2 0 ∗ 3 0∗5 0∗7
𝟎 𝟎 𝟎 𝟎
(𝟎 𝟎 𝟎 𝟎)
𝟎 𝟎 𝟎 𝟎