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.

Read explanation about matrix operations below.

Matrix Multiplication, Addition and Subtraction Calculator
  • Enter the dimension of the matrices. (Rows × Columns).
  • For multiplication, the number of columns of the first matrix must be equal to the number of rows of the second matrix, i.e. (a × b)(b × c).
  • For addition and subtraction, the dimensions of the two matrices must be the same.
  • Maximum matrix dimension for this system is 9 × 9.
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Please report any error to [email protected]
Matrix Inverse, Determinant and Adjoint Calculator
  • For a matrix to be invertible, it must be a square matrix.
  • Enter the dimension of the matrix. (Rows × Columns).
  • Maximum matrix dimension for this system is 9 × 9.
  • Result will be rounded to 3 decimal places.
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Please report any error to [email protected]


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 AB is the matrix obtained by subtracting the entries of B from the corresponding entries of A.

If A= a11 a12 a1n a21 a22 a2n am1 am2 amn and B= b11 b12 a1n b21 b22 a2n bm1 bm2 bmn

A+B= a11 + b11 a12 + b12 a1n + b1n a21 + b21 a22 + b22 a2n + b2n am1 + bm1 am2 + bm2 amn + bmn

AB= a11 b11 a12 b12 a1n b1n a21 b21 a22 b22 a2n b2n am1 bm1 am2 bm2 amn bmn

Matrices of different sizes cannot be added or subtracted.

Example:

If A= 1 2 0 -3 and B= 3 1 -1 2

A+B= 1 2 0 -3 + 3 1 -1 2 = 4 3 -1 -1

AB= 1 2 0 -3 3 1 -1 2 = -2 1 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 ABij in row i and column j of AB is given by

ABij = ai1 b1j + ai2 b2j + + air brj

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= 1 2 1 0 -3 2 and B= 3 1 0 1 -1 2 3 0 0 -2 1 1

AB= 1 2 1 0 -3 2 3 1 0 1 -1 2 3 0 0 -2 1 1 = 1 3 7 2 3 -10 -7 2

  • 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.
    AB11 = 13 + 21 + 10 = 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.
    AB12 = 11 + 22 + 12 = 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.
    AB21 = 03 + 31 + 20 = 3
  • And so on


Inverse of a Matrix

The inverse of a square matrix A is the matrix A1 such that AA1=I

Example:

If A= -3 2 5 -4 , then A1= -2 -1 -2.5 -1.5

because AA1 = -3 2 5 -4 -2 -1 -2.5 -1.5 = 1 0 0 1

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

A1 = 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.

Confused and have questions? We’ve got answers. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. If you rather get a 1:1 study help, try 30 minutes of free online tutoring with Chegg Tutors.

By Jimmy Sie

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