# IV. Matrices

```Definitions
Matrix: a set of values stored in a rectangular array.
Element: A single variable in the matrix
Row: the set of elements on a horizontal line
Column: The set of elements on a vertical line
Order: The size of a matrix, denoted (row x column). If the matrix is square, the
order can be denoted simply as (row).
Addition\Subtraction: Add and subtract elements that map to the same row and column
in the other matrix to produce a new matrix. Only matrices with identical orders can be
Multiplication by a scalar: Multiply each element by the scalar to produce a new
matrix.
Example:
1 2 5 4 8 20
48 3 4  32 12 16
7 1 5 28 4 20
Zero Matrix: A matrix with all elements equal to zero. Multiplying another matrix by a
zero matrix gives a zero matrix because each element of the non-zero matrix will be
multiplied by zero, giving a zero matrix.
Identity Matrix: A zero matrix with ones going diagonally from the upper left corner to
the bottom right. The importance of an identity matrix is that when it is multiplied by
another matrix, it returns that same matrix. In other words, if I is the identity matrix and
X is another square matrix, I*X = X
Inverse of a Matrix: As long as the determinant is not zero, the inverse of a 2x2 matrix
is
1
A
1
 d  b
a b 
1
*
= 
=
For example,

a * d  b * c  c a 
c d 
1
A
1
 7  3
 4 3
1
1  7  3
*
*
= 
=
=


4 * 7  3 * 6  6 4  10  6 4 
6 7 
The inverse matrix is used in systems of equations to find variables x, y, and z:
1
 a b c  x   j   x   a b c   j 

      
  
 d e f  y    k    y    d e f   k 
 g h i  z   l   z   g h i   l 

      
  
If the determinant is zero, either there are an infinite number of solutions or there are no
solutions.
Determinant of a square matrix: Determinants are used to find inverse matrices and to
solve linear equations. Absolute value signs around a matrix instead of parentheses
symbolize the determinant. A matrix is singular if it has a determinant of zero;
otherwise, the matrix is non-singular.
Determinant of 2x2 matrix:
a b
det(A) = |A| =
= a *d  b*c
c d
Determinant of a 3x3 matrix:
a b c
e f
d f
d e
b*
 c*
det(A) = |A| = d e f = a *
h i
g i
g h
g h i
 aei  bfg  cdh  ceg  afh  bdi
Row Reduction with Augmented matrices (Assuming the determinant is not zero):
The goal of the row reduction process is to give zeroes in all elements below the
diagonal. To do this, we may swap any two rows or add a constant multiple of any row to
another row. With combinations of these two steps, all non-zeros below the diagonal. A
reduced matrix A is referred to as A Re d . An Augmented matrix is a combination of two
matrices to create a matrix that is easier to handle. This is usually done to simplify
systems of equations. An example:
3x + 4y + z = 7
2x – y + 4z = 3
5x + 6y +z = 13
4 1  x   7 
 3

   
  2  1 4  y    3  Making the system into an augmented   5 6 1  z  13 

   
matrix by removing the (xyz) matrix and
The equals sign to combine the two
number matrices with a vertical line to
separate them:
 3
4 1 7


 2  1 4 3  Subtract 2nd row from 1st 
  5 6 1 13 


 1
5 3 4 


 2 1 4 3 
 5 6
1 13 

 1
5
3 4 


Subtract 2 times the 1 from the 2   0  11  10 5  Add 5 times the 1st to the
 5 6
1 13 

1 5
1 5
3 4 
3 4 




3rd   0  11  10 5  Add 31 (*) the 2nd to 11 (*) the 3rd   0  11  10 5 
 0 31  14 33 
0 0
464 518 



st
nd
To complete a row reduction, the second row should be divided by -11 and the last row
should be divided by 464 to give ones along the diagonal. However, before doing so, it
259
can be concluded that 464z = 518 or z =
. From the 2nd row:
232
259
 1875
-11y – 10(
) = 5 --&gt; y =
. Lastly, the 1st row gives:
1276
232
 1875
259
x + 5(
) – 3(
) = 4 --&gt; x = 14.696.
1276
232
```