Matrix

See also: Gauss-Jordan Elimination, Simultaneous Linear Equations, Geometric Linear Transformation

A matrix is a rectangular array of numbers.

The size of a matrix is its dimension, namely the number of rows and columns of the matrix.

For operations of matrices, please use the two calculators below.

To find inverse of matrix, you can also use the Gauss-Jordan Elimination method.

Please report any error to [email protected]. Thank you.



Matrix Operations

Addition and Subtraction of Matrices

If matrices A and B are of the same size,

If A =
  
 
a11a12a1n
a21a22a2n
a41a42amn
 
  
and B =
  
 
b11b12b1n
b21b22b2n
b41b42bmn
 
  
A + B =
  
 
a11+b11a12+b12a1n+b1n
a21+b21a22+b22a2n+b2n
a41+b41a42+b42amn+bmn
 
  
AB =
  
 
a11b11a12b12a1nb1n
a21b21a22b22a2nb2n
a41b41a42b42amnbmn
 
  

Matrices of different sizes cannot be added or subtracted.

Example

A =
  
 
12
0-3
 
  
and B =
  
 
31
-12
 
  
A + B =
  
 
12
0-3
 
  
+
  
 
31
-12
 
  
=
  
 
43
-1-1
 
  
AB =
  
 
12
0-3
 
  
  
 
31
-12
 
  
=
  
 
-21
1-5
 
  

Multiplication of Matrices

If A is an m × r matrix and B is an r × n matrix, the product AB is an m × n matrix whose entry from row i and column j is the sum of the products of the corresponding entries from row i of A and column j of B.

The entry (AB)ij in row i and column j of AB is given by

(AB)ij = ai1b1j + ai2b2j + ai3b3j + … + airbrj

Matrices A and B can only be multiplied if the number of columns of A is the same as the number of rows of B.

Example

A =
  
 
121
0-32
 
  
and B =
  
 
3101
-1230
0-211
 
  
AB =
  
 
121
0-32
 
  
  
 
3101
-1230
0-211
 
  
=
  
 
1372
3-10-72
 
  

Inverse of a Matrix

The inverse of a square matrix A is the matrix A-1 such that A A-1 = I

For example, if

A =
  
 
-32
5-4
 
  
, then A-1 =
  
 
-2-1
-2.5-1.5
 
  
because A A-1 =
  
 
-32
5-4
 
  
  
 
-2-1
-2.5-1.5
 
  
=
  
 
10
01
 
  

One way to get the inverse of a square matrix A is to use the following formula

A-1 = adj(A)
det(A)

If the determinant of the matrix is 0, the matrix doesn't have an inverse and it's called a singular matrix.

Another way to find the inverse of a matrix is to append an identity matrix on the right side of the matrix then use the Gauss-Jordan Elimination method to reduce the matrix to its reduced row echelon form.

By Jimmy Sie

See also: Gauss-Jordan Elimination, Simultaneous Linear Equations, Geometric Linear Transformation

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