This document provides an introduction to matrices and determinants. Some key points:
- Matrices are arrays of numbers arranged in rows and columns. Determinants are single numbers associated with square matrices.
- Methods are presented for calculating the determinants of 2x2 and 3x3 matrices, including the diagonal and cofactor methods.
- The determinant of a matrix is used to find the inverse of a matrix and in calculating the area of triangles.
- Matrix operations like multiplication and solving systems of equations can be represented using matrix notation. The inverse of a matrix allows solving systems of equations written in matrix form.
2. Matrices
• A matrix is an array of
numbers that are
arranged in rows and
columns.
• A matrix is “square” if
it has the same number
of rows as columns.
• We will consider only
2x2 and 3x3 square
matrices
0
-½
3
1
11
180
4
-¾
0
2
¼
8
-3
3. Determinants
• Every square matrix has
a determinant.
• The determinant of a
matrix is a number.
• We will consider the
determinants only of
2x2 and 3x3 matrices.
1 3
-½ 0
-3 8 ¼
2 0 -¾
4 180 11
Note the difference in the matrix
and the determinant of the
matrix!
4. Why do we need the determinant
• It is used to help us
calculate the inverse
of a matrix and it is
used when finding
the area of a triangle
11.
1
0
0
1
Identity matrix: Square matrix with 1’s on the diagonal
and zeros everywhere else
2 x 2 identity matrix
1
0
0
0
1
0
0
0
1
3 x 3 identity matrix
The identity matrix is to matrix multiplication as
___ is to regular multiplication!!!!
1
Using matrix equations
12. Multiply:
1
0
0
1
4
3
2
5
=
4
3
2
5
1
0
0
1
4
3
2
5
=
4
3
2
5
So, the identity matrix multiplied by any matrix
lets the “any” matrix keep its identity!
Mathematically, IA = A and AI = A !!
13. Inverse Matrix:
Using matrix equations
2 x 2
d
c
b
a
In words:
•Take the original matrix.
•Switch a and d.
•Change the signs of b and c.
•Multiply the new matrix by 1 over the determinant of the original matrix.
a
c
b
d
bc
ad
1
1
A
A
16. What happens when you multiply a matrix by its inverse?
1st: What happens when you multiply a number by its inverse?
7
1
7
A & B are inverses. Multiply them.
8
5
3
2
=
2
5
3
8
1
0
0
1
So, AA-1
= I
17. Why do we need to know all this? To Solve Problems!
Solve for Matrix X.
=
2
5
3
8
X
1
3
1
4
We need to “undo” the coefficient matrix. Multiply it by its INVERSE!
8
5
3
2
=
2
5
3
8
X
8
5
3
2
1
3
1
4
1
0
0
1
X =
3
4
1
1
X =
3
4
1
1
18. You can take a system of equations and write it with
matrices!!!
3x + 2y = 11
2x + y = 8
becomes
1
2
2
3
y
x
=
8
11
Coefficient
matrix
Variable
matrix
Answer matrix
Using matrix equations
19. Let A be the coefficient matrix.
Multiply both sides of the equation by the inverse of A.
8
11
8
11
8
11
1
1
1
A
y
x
A
y
x
A
A
y
x
A
1
2
2
3 -1
=
3
2
2
1
1
1
=
3
2
2
1
3
2
2
1
1
2
2
3
y
x
=
3
2
2
1
8
11
1
0
0
1
y
x
=
2
5
y
x
=
2
5
Using matrix equations
1
2
2
3
y
x
=
8
11
Example: Solve for x and y .
1
A
20. Wow!!!!
3x + 2y = 11
2x + y = 8
x = 5; y = -2
3(5) + 2(-2) = 11
2(5) + (-2) = 8
It works!!!!
Using matrix equations
Check:
