Matrix addition and subtraction
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You can add and subtract the matrix of the same size as a resulting you get the matrix of the same size.
Definition.
Addition of matrix (sum of matrix) A + B is operation of a finding of a matrix C, entry of which are equal to the sum of all corresponding entry of matrix A and B, can be defined as:
сij = aij + bij
Definition.
Subtraction of matrix (difference of matrix) A - B is operation of a finding of a matrix C, entry of which are equal to a difference of all corresponding entry of matrix A and B, can be defined as:
сij = aij - bij
Properties of matrix addition and subtraction
- Associativity: (A + B) + C = A + (B + C)
- A + Θ = Θ + A = A, where Θ - zero matrix
- A - A = Θ
- Commutativity: A + B = B + A
Examples of matrix addition and subtraction
Example 1.
| Find the sum of matrices A = |  |
4 | 2 |  |
and B = | |
3 | 1 | |
. |
| 9 | 0 | -3 | 4 |
Solution:
| A + B = | |
4 | 2 | |
+ | |
3 | 1 | |
= | |
4 + 3 | 2 + 1 | |
= | |
7 | 3 | |
| 9 | 0 | -3 | 4 | 9 + (-3) | 0 + 4 | 6 | 4 |
Example 2
| Find the difference of matrices A = | |
4 | 2 | |
and B = | |
3 | 1 | |
. |
| 9 | 0 | -3 | 4 |
Solution:
| A - B = | |
4 | 2 | |
- | |
3 | 1 | |
= | |
4 - 3 | 2 - 1 | |
= | |
1 | 1 | |
| 9 | 0 | -3 | 4 | 9 - (-3) | 0 - 4 | 12 | -4 |
Example 3
| Find the matrix С = 2A + 3B, if A = | |
4 | 2 | |
and B = | |
3 | 1 | |
. |
| 9 | 0 | -3 | 4 | |||||||
| 4 | -6 | 9 | 1 |
Solution:
| C = 2A + 3B = 2 | |
4 | 2 | |
+ 3 | |
3 | 1 | |
= | |
2·4 + 3·3 | 2·2 + 3·1 | |
= | |
17 | 7 | |
| 9 | 0 | -3 | 4 | 2·9 + 3·(-3) | 2·0 + 3·4 | 9 | 12 | ||||||||||||
| 4 | -6 | 9 | 1 | 2·4 + 3·9 | 2·(-6) + 3·1 | 35 | -9 |
MatrixMatrix Definition. Main informationSystem of linear equations - matrix formTypes of matricesMatrix scalar multiplicationAddition and subtraction of matricesMatrix multiplicationTranspose matrixElementary matrix operationsDeterminant of a matrixMinors and cofactors of a matrixInverse matrixLinearly dependent and independent rowsRank of a matrix
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