# Subtraction of Matrices

A mathematical system of subtracting one matrix from another matrix is called subtraction of matrices.

Subtraction of matrices is a mathematical operation, used to subtract one matrix from another matrix in mathematics.

Subtraction of matrices is performed by subtracting the elements of a matrix from the elements of another matrix. Therefore, the number of elements in one matrix must be equal to the number of elements in another matrix to perform the subtraction but it is not possible to perform subtraction with matrices of different order.

## Example

1.

### Subtraction of Matrices of same order

Example: 1

$A$

and

$B$

are two matrices.

$A$

is a matrix of order

$3×3$

and

$B$

is also a matrix of order

$3×3$

.

$A$
=

7
3
2

1
6
9

2
5
3

and

$B$
=

4
9
3

5
7
8

1
9
8

Assume, matrix

$B$

is subtracted from matrix

$A$

and it is written in mathematics by placing a minus sign between them.

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

Both

$A$

and

$B$

are

$3×3$

matrices. Every matrix is formed by the elements which are arranged in it in rows and columns. Subtraction of matrices means subtraction of elements.

Nine elements are there in each matrix. All nine elements are in nine different positions in every matrix. So, subtract every element of matrix

$B$

from every element in matrix

$A$

in same position to complete the process of subtraction with matrices.

Step: 1
Subtract elements in First row First column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$–7$

is an element in first row, first column in matrix

$A$

.

$4$

is an element in first row, first column in matrix

$B$

. Subtract

$4$

from

$–7$

and place the operation in first row, first column.

7   –   4

Step: 2
Subtract elements in First row second column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$3$

is an element in first row, second column in matrix

$A$

.

$–9$

is an element in first row, second column in matrix

$B$

. Subtract

$–9$

from

$3$

and place the operation in first row, second column.

7   –   4
3   –   (9)

Step: 3
Subtract elements in First row third column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$–2$

is an element in first row, third column in matrix

$A$

.

$–3$

is an element in first row, third column in matrix

$B$

. Subtract

$–3$

from

$–2$

and place the operation in first row, third column.

7   –   4
3   –   (9)
2   –   (3)

Step: 4
Subtract elements in Second row First column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$1$

is an element in second row, first column in matrix

$A$

.

$–5$

is an element in second row, first column in matrix

$B$

. Subtract

$–5$

from

$1$

and place the operation in second row, first column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)

Step: 5
Subtract elements in Second row second column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$6$

is an element in second row, second column in matrix

$A$

.

$7$

is an element in second row, second column in matrix

$B$

. Subtract

$7$

from

$6$

and place the operation in second row second column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7

Step: 6
Subtract elements in Second row third column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$9$

is an element second row third column in matrix

$A$

.

$8$

is an element in second row second column in matrix

$B$

. Subtract

$8$

from

$9$

and place the operation in second row third column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7
9   –   8

Step: 7
Subtract elements in Third row first column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$2$

is an element of matrix

$A$

in third row and first column.

$1$

is an element of matrix

$B$

in third row first column. Subtract

$1$

from

$2$

and place the operation in third row first column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7
9   –   8

2   –   1

Step: 8
Subtract elements in Third row second column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$5$

is an element of matrix

$A$

in third row second column.

$9$

is an element of matrix

$B$

in third row second column. Subtract

$9$

from

$5$

and place the operation in third row second column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7
9   –   8

2   –   1
5   –   9

Step: 9
Subtract elements in Third row third column of Matrices

$A$

and

$B$

7
3
2

1
6
9

2
5
3

–

4
9
3

5
7
8

1
9
8

$3$

is an element of matrix

$A$

in third row third column.

$8$

is an element of matrix

$B$

in third row third column. Subtract

$8$

from

$3$

and place the operation in third row third column.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7
9   –   8

2   –   1
5   –   9
3   –   8

The matrix

$B$

is subtracted from matrix

$A$

in

$9$

steps and it is time to simplify the matrix.

7   –   4
3   –   (9)
2   –   (3)

1   –   (5)
6   –   7
9   –   8

2   –   1
5   –   9
3   –   8

7   –   4
3   +   9
2   +   3

1   +   5
6   –   7
9   –   8

2   –   1
5   –   9
3   –   8

=

11
12
1

6
1
1

1
4
5

It is a matrix formed by the subtraction of matrix

$B$

from matrix

$A$

.

Example: 2
Use same process to perform subtraction of matrix

$A$

from matrix

$B$

for practicing the subtraction of matrices.

4
9
3

5
7
8

1
9
8

–

7
3
2

1
6
9

2
5
3

The nine steps of subtraction is done in only one step.

4   –   (7)
9   –   3
3   –   (2)

5   –   1
7   –   6
8   –   9

1   –   2
9   –   5
8   –   3

4   +   7
9   –   3
3   +   2

5   –   1
7   –   6
8   –   9

1   –   2
9   –   5
8   –   3

11
12
1

6
1
1

1
4
5

2.

### Subtraction of Matrices of different order

Example: 1

$C$
=

2
1
5

and

$D$
=

0
3

6
9

$C$

is a matrix of order

$1×3$

and

$D$

is another matrix of order

$2×2$

. The order of matrices

$C$

and

$D$

are different. In other words, the matrix

$C$

contains

$3$

elements in

$1$

row

$3$

columns and the matrix

$D$

contains

$4$

elements in

$2$

rows

$2$

columns.

$4$

elements cannot be subtracted from

$3$

elements and vice-versa.

Firstly, perform subtraction of matrix

$D$

from matrix

$C$

.

=

2
1
5

–

0
3

6
9

=

2  –  0
1  –  3

1. The element
$0$

is an element in first row first column of matrix

$D$

and it subtracted the element

$2$

belongs to first row first column of matrix

$C$

. The operation of the elements in first row first column of both matrices is placed in same position in matrix.

2. The element
$3$

in first row second column of matrix

$D$

subtracted the element

$1$

in first row second column of matrix

$C$

. Place the operation in same position in matrix.

3. There is no position called first row third column in matrix
$D$

but matrix

$C$

has. However, matrix

$D$

cannot subtract matrix

$C$

in this position. Therefore, subtraction of matrices is failed.

Example: 2

$E$
=

7

1

8

3

and

$F$
=

5
0

1
4

$E$

is a matrix of order

$4×1$

and

$F$

is a matrix of order

$2×2$

. Matrix

$E$

is having

$4$

elements and Matrix

$F$

is also having

$4$

elements. Four elements of one matrix can subtract four elements of another matrix.

7

1

8

3

–

5
0

1
4

Now, perform subtraction with matrices

$E$

and

$F$

.

7   –  5

1. The element
$7$

is an element belongs to first row first column in matrix

$E$

and

$5$

is an element in first row first column in matrix

$F$

. Subtract the element

$5$

from element

$7$

and place the operation in same position of the matrix.

2. There is no position, called first row second column in matrix
$E$

but the matrix

$F$

is having

$0$

as element in first row second column. Due to lack of first row second column, the matrix

$F$

cannot subtract the matrix

$E$

. Therefore, subtraction of matrix

$F$

from matrix

$E$

is not possible.

In first example, matrix

$C$

is having three elements and matrix

$D$

is having four elements. Three elements cannot subtract four elements but in the second example, subtraction is also failed even though both matrix

$E$

and

$F$

are having four elements.

Subtraction does not depend on the number of elements completely but it depends on the order of the matrices. Subtraction of two matrices is possible if they are in same order.

Therefore, first check the order of the matrices. If order of matrices is same, then perform the subtraction. Otherwise, it is not possible to perform the subtraction of matrices in mathematics.