This document defines and provides examples of common matrix operations: - A matrix is a two-dimensional array of elements arranged in rows and columns. Special types of matrices include row vectors, column vectors, square matrices, and identity matrices. - Matrix addition and subtraction are defined element-wise, where the resulting matrix has the same dimensions. - For matrix multiplication, the number of columns of the first matrix must equal the number of rows of the second. The result is a matrix with the number of rows of the first and the number of columns of the second. - Matrix inversion finds the inverse of a square matrix B such that when multiplied by the original matrix B, the result is the identity matrix.

1. Matrix Algebra

2. Matrix
A 
a11 ,, a1n
a21 ,, a2n
   
am1 ,, amn










 Aij
 
A matrix is any doubly subscripted array of
elements arranged in rows and columns.

3. Row Vector
[1 x n] matrix
   
j
n a
a
a
a
A ,
,
2
1 


4. Column Vector
 
i
m
a
a
a
a
A
2
1















[m x 1] matrix

5. Square Matrix
B 
5 4 7
3 6 1
2 1 3








Same number of rows and columns

6. Identity Matrix
I 
1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1










Square matrix with ones on the
diagonal and zeros elsewhere.

7. Transpose Matrix
A' 
a11 a21 ,, am1
a12 a22 ,, am2
    
a1n a2n ,, amn










Rows become columns and
columns become rows

8. Matrix Addition and
Subtraction
A new matrix C may be defined as the
additive combination of matrices A and B
where: C = A + B
is defined by:
Cij
   Aij
   Bij
 
Note: all three matrices are of the same dimension

9. Addition
A 
a11 a12
a21 a22






B 
b11 b12
b21 b22






C 
a11  b11 a12  b12
a21  b21 a22  b22






I
f
and
then

10. Matrix Addition Example
A  B 
3 4
5 6







1 2
3 4







4 6
8 10






 C

11. Matrix Subtraction
C = A - B
Is defined by
Cij
   Aij
   Bij
 

12. Matrix Multiplication
Matrices A and B have these dimensions:
[r x c] and [s x d]

13. Matrix Multiplication
Matrices A and B can be multiplied if:
[r x c] and [s x d]
c = s

14. Matrix Multiplication
The resulting matrix will have the dimensions:
[r x c] and [s x d]
r x d

15. Computation: A x B = C
A 
a11 a12
a21 a22






B 
b11 b12 b13
b21 b22 b23



















23
22
13
21
22
22
12
21
21
22
11
21
23
12
13
11
22
12
12
11
21
12
11
11
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
b
a
C
[2 x 2]
[2 x 3]
[2 x 3]

16. Computation: A x B = C
A 
2 3
1 1
1 0








and B 
1 1 1
1 0 2






[3 x 2] [2 x 3]
A and B can be multiplied








































1
1
1
3
1
2
8
2
5
1
2
*
0
1
*
1
1
0
*
0
1
*
1
1
1
*
0
1
*
1
3
2
*
1
1
*
1
1
0
*
1
1
*
1
2
1
*
1
1
*
1
8
2
*
3
1
*
2
2
0
*
3
1
*
2
5
1
*
3
1
*
2
C
[3 x 3]

17. Computation: A x B = C








































1
1
1
3
1
2
8
2
5
1
2
*
0
1
*
1
1
0
*
0
1
*
1
1
1
*
0
1
*
1
3
2
*
1
1
*
1
1
0
*
1
1
*
1
2
1
*
1
1
*
1
8
2
*
3
1
*
2
2
0
*
3
1
*
2
5
1
*
3
1
*
2
C
A 
2 3
1 1
1 0








and B 
1 1 1
1 0 2






[3 x 2] [2 x 3]
[3 x 3]
Result is 3 x 3

18. Matrix Inversion
B1
B  BB1
 I
Like a reciprocal
in scalar math
Like the number one
in scalar math