Download Matrix Vocabulary

Matrix Vocabulary
Dimensions of a matrix- The dimensions are determined by the number of rows and
columns.
2
4
1X2
One row, 2 columns
1 3 0 


 0 0 1
2X3
2 rows, 3 columns
Entries- The numbers inside the matrix.
 a11

 a21
a
 31


1 1 1 1


 2 2 2 2
 3 3 3 3


3X4
3 rows, 4 columns
a13 

a23 
a33 
a12
a22
a32
a21 means the entry is in the second row, first column.
a34 means the entry is in the third row, fourth column.
Square Matrix- There is an equal number of rows and columns
 a11 a12 

 2X2
 a21 a22 
Rectangular Matrix- The number of rows is not equal to the number of columns.
 a11 a12 a13 

 2X3
 a21 a22 a23 
Column vector- One column of entries
1
 
2 
 
 
m
Row vector- A row of entries.
1 2
mX1
m
1Xm
Addition of matrices
Definition:
a b   e


c d  g
f  ae b f 


h  c  g d  h
Example:
 1 4   3 5  1  (3) 4  (5) 




67 
8 6  0 7   8  0
 2 1


 8 13 
*In order to add or subtract two matrices, their dimensions must be the same.
Subtraction of matrices
Definition:
a b  e


c d g
f   ae b f 


h  c  g d  h
Example:
 1 4   3 5  1  (3) 4  (5) 




67 
8 6  0 7   8  0
4 9 


 8 1
*In order to add or subtract two matrices, their dimensions must be the same
*In order to multiply two matrices, the # of columns of the first matrix must equal the
number of rows of the second.
To determine if it is possible to multiply two matrices check the number of columns of
the first matrix with the rows of the second (The inside numbers):
2 X 5 and 5 X 3 →
4 X 6 and 4 X 6 →
Yes
No
The answer will be determined by the rows of the first matrix and the columns of the
second (The outside numbers):
3 X 5 and 5 X 2 →
2 X 4 and 4 X 5 →
3X2
2X5
Multiplication of Matrices
Definition:
a b   e

 x
c d  g
f   ae  bg

h   ce  dg
af  bh 

cf  dh 
Example:
 1 2   3 5   1(3)  2(5) 1(5)  2(1) 

 x


 3 4   5 1   3(3)  4(5) 3(5)  4(1) 
 1 2   3 5   7 7 

 x


 3 4   5 1   11 19 
Identity Matrix
The identity matrix, I, is similar to the 1 in standard multiplication. Remember the
identity property: Anything times 1 is itself.
 1 2  1 0   1 2 




 3 1 0 1   3 1
1
1X1
1 0


0 1
2X2
1 0 0


0 1 0
0 0 1


3X3
AI = A

The identity matrix is a matrix consisting of 1’s and
0’s. The ones are found along the diagonal of the
matrix starting in the top right corner.

The identity matrix is always a square matrix.

The size is determined by the number of columns of
the original matrix. A 2 X 3 matrix would have a
3 X 3 identity.
Inverse Matrix
1
The inverse matrix, A , is the matrix that will produce an identity matrix when
multiplied by the original matrix. It can be used to solve systems of equations.


Rectangular matrices do not have inverses.
Not all square matrices have inverses.
If we have a matrix equation Ax  B you can solve by multiplying both sides of the
equation by A1 , A1 Ax  A1 B .
Let’s say we have the system of equations:
x  2y  5
3 x  y  2
 1 2  x   5 
We could write the matrix equation: 
    
 3 1  y   2 
1 2
 0.2 0.4 
1
So A  
 and A  
 and
3 1
 0.6 0.2 
 x   0.2 0.4  5 
 
 
 y   0.6 0.2  2 
 x   1.8 
From this multiplication we get    

 y   3.4 