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MATRICES
MATRIX Multiplication
Warm-up
Subtract (don’t forget to KCC):
é 0 3 ù é -3 5 ù é
ê
ú ê
ú ê
ê 5 -5 ú - ê 0 7 ú =ê
ê 4 -1 ú ê -6 -2 ú êë
ë
û ë
û
ù
3 -2
ú
5 -12 ú
10 1 ú
û
Matrix Multiplication


Matrix Multiplication is NOT
Commutative! Order matters!
You can multiply matrices only if the
number of columns in the first matrix
equals the number of rows in the second
matrix.
2 columns
3ù
é2
ê -5 6 ú · é 1
ê
ú ê3
êë 9 -7 úû ë
-2 0 ù
4 -5 úû
2 rows
Matrix Multiplication

Take the numbers in the first row of
matrix #1. Multiply each number by its
corresponding number in the first
column of matrix #2. Total these
products.
3ù
é2
ê -5 6 ú · é 1
ê
ú ê3
êë 9 -7 úû ë
-2 0 ù
4 -5 úû
2i1+ 3i3 = 11
The result, 11, goes in
row 1, column 1 of the
answer. Repeat with
row 1, column 2; row 1
column 3; row 2,
column 1; ...
Matrix Multiplication

Notice the dimensions of the matrices and
their product.
3ù
é2
ê -5 6 ú · é 1
ê
ú êë 3
êë 9 -7 úû
3x2
__
8 -15 ù
é 11
-2 0 ù ê
ú
=
13
34
-30
ú
4 -5 úû ê
êë -12 -46 35 úû
2 x__
3
3 x__
3
__
Matrix Multiplication

Another example:
2 1
 9 0    5  

  2 
10 5  
3x2
2x1
 8 
 45


 60 
3x1
Matrix Determinants
 A Determinant is a real number associated
with a matrix. Only SQUARE matrices
have a determinant.
 The symbol for a determinant can be the
phrase “det” in front of a matrix variable,
det(A); or vertical bars around
a matrix, |A| or 3 -1 .
2
4
Matrix Determinants
To find the determinant of a 2 x 2 matrix,
multiply diagonal #1 and subtract the product
of diagonal #2.
Diagonal 2 = -2
3 1

2 4
Diagonal 1 = 12
12  (2)  14
Matrix Determinants
To find the determinant of a 3 x 3 matrix, first
recopy the first two columns. Then do 6
diagonal products.
18
60 16
5
2
6 5
2
2
-1 4 2
-1
-3
3
4 -3
3
-20 -24
36
Matrix Determinants
The determinant of the matrix is the sum of
the downwards products minus the sum of the
upwards products.
18
60
6 5
16
5
2
2
2
-1 4 2
-1
-3
3
4 -3
3
-20 -24
= (-8) - (94) = -102
36
Identity Matrices

An identity matrix is a square matrix that
has 1’s along the main diagonal and 0’s
everywhere else.
é1 0 0 ù
ê0 1 0 ú
ê
ú
êë 0 0 1 úû

é1 0 ù
ê0 1 ú
ë
û
When you multiply a matrix by the
identity matrix, you get the original
matrix.
Inverse Matrices
 When you multiply a matrix and its
inverse, you get the identity matrix.
é 3 -1ù é 2 1 ù é 1 0 ù
ê -5 2 úiê 5 3ú = ê 0 1 ú
ë
ûë
û ë
û
Inverse Matrices
 Not all matrices have an inverse!
 To find the inverse of a 2 x 2 matrix,
first find the determinant.
a) If the determinant = 0, the inverse does
not exist!
 The inverse of a 2 x 2 matrix is the
reciprocal of the determinant times the
matrix with the main diagonal swapped
and the other terms multiplied by -1.
Inverse Matrices
3 -1ù
é
Example 1: A = ê
ú
-5
2
ë
û
det(A) = 6 - (5) = 1
1 é2 1ù é2 1ù
A = ê
=ê
ú
1 ë 5 3û ë 5 3úû
-1
Inverse Matrices
Example 2:
é -2 -2 ù
B=ê
ú
5
4
ë
û
det(B) = (-8) - (-10) = 2
1é 4 2 ù é 2
B = ê
=ê 5
ú
2 ë -5 -2 û ë - 2
-1
1ù
-1úû