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.
Matrix Addition, Subtraction, and Multiplication
- Enter the dimension of the matrices. (Rows x Columns).
- For multiplication, the number of columns of the first matrix must be equal to the number of rows of the second matrix, ie (a x b)(b x c).
- For addition and subtraction, the dimension of the two matrices must be the same.
- Maximum matrix dimension for this system is 9x9.
loading . . .
Calculating . . .
Inverse of Matrix
- Enter the dimension of the matrix. (Rows x Columns).
- Maximum matrix dimension for this system is 9x9.
- Result will be rounded to 3 decimal places.
loading . . .
Calculating . . .
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,
- the sum A + B is the matrix obtained by adding the entries of B to the corresponding entries of A.
- the difference A − B is the matrix obtained by subtracting the entries of B from the corresponding entries of A.
If A = | | and B = | |
A + B = | |
A − B = | |
Matrices of different sizes cannot be added or subtracted.
Example
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} = a_{i1}b_{1j} + a_{i2}b_{2j} + a_{i3}b_{3j} + … + a_{ir}b_{rj}
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
- The element at row 1 and column 1 of AB is obtained from summing up the product of corresponding entries of row 1 of A and column 1 of B, i.e.
(AB)_{11} = (1)(3) + (2)(-1) + (1)(0) = 1
- The element at row 1 and column 2 of AB is obtained from summing up the product of corresponding entries of row 1 of A and column 2 of B, i.e.
(AB)_{12} = (1)(1) + (2)(2) + (1)(-2) = 3
- The element at row 2 and column 1 of AB is obtained from summing up the product of corresponding entries of row 2 of A and column 1 of B, i.e.
(AB)_{21} = (0)(3) + (-3)(-1) + (2)(0) = 3
- And so on
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
One way to get the inverse of a square matrix A is to use the following formula
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