Matrix Calculator
Matrix calculator is a free-to-use online mathematics tool which can calculate various matrix operations.You can calculate the transpose, inverse, determinant, size, power, root, addition, subtraction, multiplication, division, and scalar operations of a matrix.
With the help of up,down, left and right arrows, adjust the rows and columns of the matrices. After that if you want to perform the multi matrices operations then you can directly do that by clicking on operations shown below and if you want to perform the single matrix operations then you should select the matrix either matrixA or matrixB according to your choice.
Matrix Calculator
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Transpose of a Matrix
The transpose of a matrix is a matrix which is obtained by rotating the matrix elements around the diagonal.This means that if the position of an element of a matrix is (i,j) then its position in the transposed matrix will be (j,i).
Hence, interchanging the positions of row to column and column to row gives the transpose of a matrix.
For example:
Inorder to find the transpose of the above matrix we can just change the row to col and col to row like this.
2↔4
3↔7
6↔8
you can observer that the diagonal elements are unchanged.
Hence, interchanging the positions of row to column and column to row gives the transpose of a matrix.
For example:
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 5 |
Inorder to find the transpose of the above matrix we can just change the row to col and col to row like this.
| 1 | 4 | 7 |
| 2 | 5 | 8 |
| 3 | 6 | 5 |
2↔4
3↔7
6↔8
you can observer that the diagonal elements are unchanged.
Determinant of a Matrix
Determinant of a matrix is a dimention less value which is calculated from the elements of a square matrix. The determinant of a matrix can be used to solve the system of linear equations or find the inverse matrix. Determinant of a matrix have many more applications in the field of mathematics.We represent the determinant in this Symbolic form |A| .
If the determinant of a matrix is zero then the inverse of such a matrix donot exist. Inorder to calculate a determinant the matrix must be a square matrix.
For example:
we can find the determinant of a 2 x 2 matrix as:
If the determinant of a matrix is zero then the inverse of such a matrix donot exist. Inorder to calculate a determinant the matrix must be a square matrix.
For example:
we can find the determinant of a 2 x 2 matrix as:
A=
= a*d - b*c
Similarly, for a 3 x 3 matrix, we can find the determinant by reducing it into 2 x 2 matrix as follows:| a | b |
| c | d |
B=
| a | b | c |
| d | e | f |
| g | h | i |
=a*
| e | f |
| h | i |
-b*
| d | f |
| g | i |
+c*
| d | e |
| g | h |
=a*(ei-fh)-b*(di-gf)+c*(dh-eg) =aei+bfg+cdh-ceg-bdi-afh
You are free to choose any row or column but you should choose a row or a column full. It becomes so easy if any element is zero in that row or column reducing the time to calculate.
For the sign we choose above you can remember it as:
if the position of element is (i,j) then the sign will be .
In above example the position of a is (1,1) so the sign is positive. while the position of b is (1,2) so the sign is negative and so on.
For the sign we choose above you can remember it as:
if the position of element is (i,j) then the sign will be .
In above example the position of a is (1,1) so the sign is positive. while the position of b is (1,2) so the sign is negative and so on.
Inverse of a Matrix
Inverse of a matrix A is an another matrix of same size which when multiplied by the matrix A gives the identity matrix. This means that:
Identity matrix (I) is a square matrix whose diagonal elements are 1 and all other elements are 0. Its determinant is always 1.
For the existence of the inverse of a particular matrix, its determinant should not be zero .
We can find the inverse of a matrix using the formula as:
where adj(A) is the adjoint of the matrix A.
Identity matrix (I) is a square matrix whose diagonal elements are 1 and all other elements are 0. Its determinant is always 1.
For the existence of the inverse of a particular matrix, its determinant should not be zero .
We can find the inverse of a matrix using the formula as:
where adj(A) is the adjoint of the matrix A.
Adjoint of a matrix is the transpose of a cofactor matrix.
Cofactor Matrix is the matrix composed up of cofactors. We can calculate the cofactor as follows:
Suppose, the given matrix is:
Cofactor Matrix is the matrix composed up of cofactors. We can calculate the cofactor as follows:
Suppose, the given matrix is:
A=
For every elements there exist Minor. Minor is obtained by removing the row and column of the chosen element.| a11 | a12 | a13 |
| b21 | b22 | b23 |
| c31 | c32 | c33 |
The minor of a11 is
| b22 | b23 |
| c32 | c33 |
The minor of c32 is
| a11 | a13 |
| b21 | b23 |
Cofactor of an element of a matrix is obtained by multiplying the minor with
Cofactor is denoted by capital letter of the element. if m11 is the minor then M11 is the cofactor.
Cofactor is denoted by capital letter of the element. if m11 is the minor then M11 is the cofactor.
Cofactor of a11 = A11 = *
| a11 | a13 |
| b21 | b23 |
Matrix of cofactors =
| A11 | A12 | A13 |
| B21 | B22 | B23 |
| C31 | C32 | C33 |
Now the transpose of this cofactor matrix is the ajdoint of given matrix.
Adjoint of A =
| A11 | B21 | C31 |
| A12 | B22 | C32 |
| A13 | B23 | C33 |
Hence, the inverse of a matrix is calculated as:
| A11 | B21 | C31 |
| A12 | B22 | C32 |
| A13 | B23 | C33 |
In summary:
1. Find the determinant of the matrix A. If the determinant is not zero then proceed forward else stop.
2. Find the cofactor matrix of the matrix A.
3. Find the adjoint of the matrix A.
4. Divide the adjoint of the matrix A by the determinant of the matrix A.
2. Find the cofactor matrix of the matrix A.
3. Find the adjoint of the matrix A.
4. Divide the adjoint of the matrix A by the determinant of the matrix A.
Size of a Matrix
The size of a matrix is the number of rows and number of comumns elements are occupying. Generally, the size of a matrix is represented as m x n (row x column).
For example:
Since, there are 4 rows and 3 columns so the size is 4 x 3.
For example:
| 1 | 2 | 3 |
| 4 | 5 | 6 |
| 7 | 8 | 9 |
| 4 | 5 | 6 |
