Overview
Yesterday: systems of equations
and rowreduction.

  

1
2
1
x
12

  

For example, the system 4
3 −11  y  =  13
5 −1 −28
z
−17
has
the
same
solutions
as
the
reduced
system

   
−2
x
1 0 −5

   
3  y  =  7.
0 1
0
z
0 0
0
Math 311-102
Harold P. Boas
boas@tamu.edu
Today: matrix algebra (sums, products, inverses). For
example,
find a matrix
M such thatthe matrix product

 
1
2
1
1 0 −5

 

M 4
3 −11 = 0 1
3.
5 −1 −28
0 0
0
Math 311-102
June 2, 2005: slide #1
Math 311-102
Componentwise operations
Matrix multiplication
Sums and differences of matrices are computed
componentwise.
Example.
Ã
10
30
20
40
!
−
Ã
1
3
2
4
!
=
Ã
9
27
The product of a matrix with a column vector is a new vector
obtained
by
 with the rows of the matrix:

 taking

dot products

1 2 3
1
321



 
4 5 6  10  = 654.
987
7 8 9
100
!
18
.
36
The product of a matrix with another matrix is computed
similarly by letting
act oneachcolumn of the
 the firstmatrix


1 0 2
201 1005
1
5



 
second matrix: 0 3 4  10
50  = 430 2150.
5 6 0
65
325
100 500
Multiplication of a matrix by a scalar is computed
componentwise.
Example. 10
Ã
1
3
2
4
!
=
Ã
10
30
!
20
.
40
The product of matrices makes sense only if the rows of the
first matrix have the same number of entries as the columns of
the second matrix.
The product of a matrix with a matrix, however, is not
computed componentwise.
Math 311-102
June 2, 2005: slide #2
June 2, 2005: slide #3
Math 311-102
June 2, 2005: slide #4
Multiplication is not commutative!
Example.
Ã
!Ã
1 0
1
0
−1 0
Ã
!Ã
1 1
1
0 0
−1
1
0
!
0
0
!
=
Ã
1
−1
=
Ã
0
0
Identity and inverses


1 0 0


For 3 × 3 matrices, the identity matrix I = 0 1 0 has the
0 0 1
property that I A = A and AI = A for every 3 × 3 matrix A (and
similarly for square matrices of other sizes).
!
1
, but
−1
!
0
.
0
Two matrices A and B are inverses if AB = I and BA = I.
The order of multiplication matters: AB 6= BA.
For 2 × 2 matrices, there is a simple rule for the inverse of a
matrix:
Ã
!−1
Ã
!
1
a b
d −b
=
.
ad − bc −c
c d
a
On the other hand, matrix multiplication is associative:
A(BC) = (AB)C.
Notice in the above example that the product of two non-zero
matrices can be the zero matrix. In particular, only certain
matrices have multiplicative inverses.
Math 311-102
June 2, 2005: slide #5
Computing inverses in general
Method: write the matrix next to an identity matrix and do row
operations on both matrices simultaneously. When the initial
matrix has been turned into the identity matrix, the identity
matrix will have been turned into the inverse matrix. Example:


 ¯
1 0 −1 ¯¯ 1 0 0


 ¯
row reduce →
0 ¯ 0 1 0
 2 1
¯
¯
0 0 1
−2 0
1

 
 ¯

 ¯
1 0 0
1 0 0 ¯¯ −1 0 −1
1 0 −1 ¯¯

 
 ¯

 ¯
2
2 ¯−2 1 0 → 0 1 0 ¯ 2 1
0 1
¯
¯
¯
¯
−2 0 −1
2 0 1
0 0 1
0 0 −1

−1 

1 0 −1
−1 0 −1




Conclusion:  2 1
0 =  2 1
2.
−2 0
1
−2 0 −1
Math 311-102
June 2, 2005: slide #7
This inverse exists if and only if the determinant (ad − bc) 6= 0.
Math 311-102
June 2, 2005: slide #6