Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Basic Matrix Operations

1. Let A = (aij) be an m × n matrix:

2. The transpose of A is the n × m matrix

The rows of A are the columns of At. For example, if

then

Note that (At)t = A.

3. If c is a scalar (i.e. a real or complex number) and A = (aij )is an m × n matrix, then cA is the m × n
matrix with entries caij . So if A is given by (1) and c = 2, then

4. If A = (aij) and B = (bij) are m × n matrices, then the sum A + B is the m × n matrix with entries
aij + bij . So if A is given by (1) and

then

5. If A is an m × n matrix and B is n × k, then the product AB is the m × k matrix whose ijth entry is
the dot product of the ith row of A and the jth column of B. For example, let

Since A is 2 × 3 and B 3 × 3, the product AB will be a 2 × 3 matrix. In particular,

6. The m×m identity matrix Im is the m×m matrix with main diagonal entries 1 and 0’s everywhere else.
So, for example,

7. An m × m matrix A is invertible, or nonsingular if there is an m × m matrix A−1 such that

The matrix A−1 is called the inverse of A.

8. The determinant of the 2 × 2 matrix

is

If det(A) ≠ 0, then A is invertible, and its inverse is

Since the matrix

has det(A) = 6, it is invertible and