Matrix Addition Calculator

1. Choose the operation, add or subtract.
2. Choose the matrix size, 2 by 2 or 3 by 3.
3. Enter Matrix A and Matrix B row by row.
4. Read the result matrix, shown row by row.

Here is what each result means:

Matrix addition combines two matrices of the same size by adding the numbers in matching positions. The entry in row i, column j of the result is simply the sum of the two entries in that spot. Subtraction works the same way, taking the difference instead. So the two matrices must have exactly the same shape.

Unlike matrix multiplication, addition is straightforward and follows the familiar rules of ordinary arithmetic. It is commutative, so A plus B equals B plus A, and it is used whenever you combine two data grids, images or transformations term by term.

Because each position is handled on its own, the two matrices never mix across cells the way they do in multiplication. That keeps the rules simple: the shapes must match, the answer has the same shape, and every entry depends only on the two numbers in that one spot. Subtraction is the same operation with the second matrix negated first.

It pairs up the entries in the same position and adds or subtracts them.

1. Line up the two matrices by position.
2. Add or subtract each matching pair of entries.
3. Place each result in the same spot in the answer.

To combine matrices by the row-by-column rule instead, use the matrix multiplication calculator.

Add the 2x2 matrices with rows 1, 2 and 3, 4, and 5, 6 and 7, 8.

Calculation: add matching entries: 1 + 5 = 6, 2 + 6 = 8, and so on. The result is [6, 8 ; 10, 12]. Subtracting instead would give [-4, -4 ; -4, -4].

A few simple rules govern adding matrices.

These make addition much simpler than multiplication, which is not commutative and has stricter size rules.

The two operations are very different in rules and use.

Use addition to combine matrices term by term, and multiplication to compose transformations.

The Operation

Adding raises matching entries; subtracting takes their difference, which can be negative.

The Matrix Sizes

The two matrices must be the same shape. Different sizes cannot be added.

The Order for Subtraction

A minus B is not the same as B minus A, since subtraction reverses the sign.

Linear Algebra Homework

Add or subtract matrices quickly and check each entry.

Combining Data Grids

Merge two tables of the same shape, cell by cell.

Images and Transforms

Blend two grids of values, as in simple image or signal work.

1. Adding Different Sizes

Matrices must be the same shape to add. Different sizes cannot be combined.

2. Adding Across Positions

Only entries in the same row and column are combined, not any others.

3. Reversing Subtraction

A minus B differs from B minus A. Keep the order you intend.

4. Mixing Up Rows and Columns

Enter each matrix row by row so entries land in the right spots.

5. Sign Slips

Subtraction can give negative entries. That is correct, not an error.

The arithmetic is exact; only the displayed decimals are rounded.

Add the numbers in matching positions. The entry in row i, column j of the sum is the sum of the two entries there. Both matrices must be the same size.

No. Matrix addition requires the two matrices to have exactly the same shape, so every entry has a partner to add. Different sizes cannot be added.

Yes. A plus B equals B plus A, just like adding numbers. Subtraction, however, is not, since A minus B differs from B minus A.

Subtract the entries in matching positions. The entry in row i, column j of the difference is the first matrix entry minus the second. The order matters.

It is a matrix of all zeros. Adding it to any matrix of the same size leaves that matrix unchanged, so it plays the role of zero in addition.

Addition combines entries in the same position and needs equal sizes. Multiplication uses the row-by-column rule, has stricter size rules, and depends on the order.

Yes. Matrices can hold any real numbers, and subtraction often produces negative entries. The tool handles decimals and negatives.

It combines two grids of numbers term by term, which appears in data tables, image processing, and adding transformations or forces represented as matrices.

No. The calculation runs in your browser and nothing you enter is stored or sent anywhere, unless you choose Save, which keeps the result only on this device.

• Matrix addition (Wikipedia).
• Adding and subtracting matrices (Maths Is Fun).
• Matrix (mathematics) (Wikipedia).