ODE Lecture Notes
Section 7.2
Page 1 of 8
Section 7.2: Review of Matrices
Big Idea: x
Big Skill: x
A matrix is a rectangular array of numbers (called elements) arranged in m rows and n columns:
a1n 
 a11 a12
a
a22
a2 n 
A   21




amn 
 am1 am 2
aij is used to denote the element in row i and column j.
(aij) is used to denote the matrix whose generic element is aij.
The transpose of a matrix is a new matrix that is formed by interchanging the rows and columns
of a given matrix. The transpose is denoted with a superscript T:
1 2 3
1 4 7 


T
if A  4 5 6 , then A   2 5 8  ; if A = (aij), then AT = (aji).




7 8 9 
 3 6 9 
The conjugate of a matrix is a new matrix that is formed by taking the conjugate of every
element of a given matrix. The conjugate is denoted with a bar over the matrix name.
 1  i 2  3i 
 1  i 2  3i 
if A  
, then A  

 ; if A   aij  , then A   aij  .
 4  5i 6  7i 
 4  5i 6  7i 
The adjoint of a matrix is the transpose of the conjugate of a given matrix. The adjoint is
denoted with a superscript asterisk:
 1  i 2  3i 
 1  i 4  5i 
if A  
, then A*  

.
 4  5i 6  7i 
 2  3i 6  7i 
The two most useful matrices in this chapter will be square with n rows and n columns (these are
called matrices of order n), and n  1 column matrices with n rows and one column (used to
represent vectors).
T
1 
Note: the transpose of a column vector is a row vector:  2   1 2 3
 3 
ODE Lecture Notes
Section 7.2
Page 2 of 8
Properties of Matrices
1. Equality: Two matrices A and B are equal only if all corresponding elements are equal;
i.e., aij = bij for all i and j.
2. Zero: The matrix with zero for every element is denoted as 0.
3. Addition: The sum of two m  n matrices A and B is computed by adding corresponding
elements:
A  B   aij    bij    aij  bij 
1 2  5 6  6 8 
3 4   7 8   10 12 

 
 

a. Matrix addition is commutative and associative:
A  B  B  A ; A   B  C   A  B   C
4. Multiplication by a number: The product of a complex number and a matrix is
computed by multiplying every element by the number:
 A    aij    aij 
1 2  2  i 4  2i 


3 4 6  3i 8  4i 
a. Multiplication by a constant obeys the distributive laws
  A  B    A   B ;     A   A   A
b. The negative of a matrix is negative one times the matrix:
A   1 A
2  i 
5. Subtraction: The difference of two m  n matrices A and B is computed by subtracting
corresponding elements:
A  B   aij    bij    aij  bij 
1 2 5 6  4 4
1 1
3 4  7 8    4 4   4  1 1 . Note: the last step is not required, but it

 
 



illustrates a simplifying technique that is sometimes used.
6. Multiplication of matrices: Matrix multiplication is only defined when the number of
columns in the first factor equals the number of rows in the second factor.
a. The product of an m  p and a p  n matrix is an m  n matrix.
b. If AB  C , then element cij is computed by taking the sum of the products of
corresponding elements from row i in matrix A and column j of matrix B:
p
cij   aik bkj
k 1
c. Matrix multiplication is associative:
 AB C  A  BC
ODE Lecture Notes
Section 7.2
Page 3 of 8
d. Matrix multiplication is distributive:
A  B  C  AB  AC
e. Matrix multiplication is usually NOT commutative:
AB  BA (usually)
1 2 1 
 2 1 1


f. Practice: Given A  0 2 1 and B  1 1 0  , compute AB and BA .




 2 1 1 
 2 1 1 
ODE Lecture Notes
Section 7.2
Page 4 of 8
7. Multiplication of vectors:
n
a. The dot product: xT y   xi yi
i 1
1 
4


b. Practice: compute the dot product of x  2 and y   5  .
 
 
 3 
 6 
c. Properties of the dot product:
xT y  yT x ; xT  y  z   xT y  xT z ;  xT  y   xT y  xT  y 
n
d. The scalar (inner) product:  x, y    xi yi  xT y
i 1
 i 
2  i 


e. Practice: Compute  x, y  ,  x, x  , and x x for x  2 and y   i  .




1  i 
 3 
T
f. Note: the magnitude, or length, of a vector x can be represented with the inner
product: x   x, x  .
g. Note: Two vectors are orthogonal if  x, y   0 ; the (orthogonal) unit vectors are
1 
0 
0




i  0  , j  1  , and k  0  .
0 
0 
1 
ODE Lecture Notes
Section 7.2
Page 5 of 8
h. Properties of the scalar product:
 x, y    y, x  ,  x, y  z    x, y    y, z  ,  x, y     x, y  ,  x, y     x, y 
8. Identity: The multiplicative identity for matrices, denoted by I, is the square matrix with
one in every diagonal element and zero elsewhere. The diagonal runs from the top left to
bottom right, only.
0
1 0
0 1
0 

I




1
0 0
a. Multiplication by the identity matrix is commutative for square matrices:
AI  IA  A .
1 2 1 
b. Practice: Compute AI and IA for A  0 2 1 .


 2 1 1 
ODE Lecture Notes
Section 7.2
Page 6 of 8
9. Inverse: For some square matrices, there exists another unique matrix such that their
product is the identity. Such matrices are said to be nonsingular, or invertible. Matrices
that are not invertible are called noninvertible or singular. The “other unique matrix” is
called the (multiplicative) inverse, and is denoted with a superscript -1:
If A is invertible, then AA-1 = A-1A = I.
a. The determinant is a quantity that can be computed for any square matrix, and
whose value determines if a matrix is singular or not. A zero determinant means
the matrix is noninvertible. The determinant of a matrix A is denoted as:
det(A) = |A|.
b. The determinant of any square matrix can be computed by taking the sum of the
products of all the elements and their cofactors from any one row or column of
the matrix.
c. The cofactor Cij of a given element aij is the product of the minor Mij of the
element and an appropriate factor of -1:
Cij = (-1)i+jMij .
d. The minor Mij of an element aij is the determinant of the matrix obtained by
eliminating the ith row and jth column from the original matrix.
e. Thus, a determinant can be computed using a chain of smaller determinants. It is
useful to know the following formula for the last step in the chain:
a11 a12
a b
 ad  bc
 a11a22  a12 a21 
c d
a21 a22
f. Practice: Compute the determinant and the cofactor matrix of
 1 1 2 0 
 1 2 4 2 

A
1 0 1 3


 2 2 0 1
ODE Lecture Notes
g. If B = A-1, then bij 
Section 7.2
C ji
det  A 
Page 7 of 8
, where (Cji) is the transpose of the cofactor matrix of
A.
h. Practice: Compute the inverse of the above matrix A using this formula, and
show it is the inverse by computing AA-1.
ODE Lecture Notes
Section 7.2
Page 8 of 8
i. This formula is extremely inefficient for computing the inverse. The preferred
method for computing the inverse of a matrix A is to form the augmented matrix
A | I, and then perform elementary row operations on the augmented matrix until
A is transformed to the identity matrix. That will leave I transformed to A-1. This
process is called row reduction or Gaussian elimination.
j. The three elementary row operations are:
i. Interchange any two rows.
ii. Multiply any row by a nonzero number.
iii. Add any multiple of one row to another row.
k. Practice: Compute A-1 again using Gaussian elimination.
10. Matrix Functions: Are matrices with functions for the elements. They can be integrated
and differentiated element-by-element…