Matrix: Definition, Notation, Addition, Subtraction, Transpose, Examples

Definitions of Matrix

Matrix is numbers that are arranged into row and column. If there is more than one matrix called matrices. There is no rule about the number of row and column. The number of row and column is called dimensions.

Notation of Matrix

If there are a, b, c, d, e, f, … as any numbers, then matrix X is

Notation of Matrix

Its dimension is 3×2. Dimension tells (the number of row) x (the number of columns).

In general form,  

General Form of Matrix

Note: am,n called as element of matrix.

Addition of Matrix

Addition of matrix has same operation with addition of algebra. It is just adding the number/element of matrix one by one as each position. The important thing is the matrices must have same dimension.

For example, if there are matrix A and B, then

Addition of Matrix

Note: A+B = B+A (the other rule in addition of algebra are also valid in addition of matrices)

Subtraction of Matrix

Subtraction of matrix has same rule with addition. It has same operation with subtraction of algebra. Besides that, the matrices are also must have same dimension. Matrices operated one by one elements.

For example, if there are matrix A and B, then

Subtraction of Matrix

Note: A-B ≠ B-A (the other rule in subtraction of algebra are also valid in subtraction of matrices)

Transpose a matrix

Transpose matrix means moving position of the elements. The numbers in a row move to be column and vice versa. It has symbol “T” in top right-hand corner of the matrix’s name.

For example:

Transpose of Matrix

Note: be careful, transpose change the matrix dimension. In example, it is 3×2 changed to be 2×3.


  1. Determine the dimension of the matrices and the dimension after transpose.
Matrix Examples 1
MatrixDimension before transposeDimension after transpose

2. Determine the result

Matrix Examples 2a
Matrix Examples 2b

3. Determine the result

Matrix Examples 3a
Matrix Examples 3b

4. Determine the sum of x and y.

Matrix Examples 4a
Matrix Examples 4b


  • 2 + y = 5 → y = 5 – 2 = 3
  • x + 3 = -2 → x = -2 – 3 = -5

so, x = -5 and y = 3

5. there are matrix A and B

Matrix Examples 5a

Then, determine transpose of A – 2B.

  • First, determine the matrix of A – B.

Let the result is matrix C.

Matrix Examples 5b
  • Second step is transpose
Matrix Examples 5c

Read Also

Matrix Multiplication

Determinant of Matrix

Invers of a Matrix

Solving Systems of Linear Equations Using Matrix