The Order of a Matrix and How to Add Matrices
Rows before columns, and why that matters. When two matrices can be added, how to add them entry by entry, what scalar multiplication does, and why a 2 x 3 and a 3 x 2 matrix can never be summed.
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Before you can do anything with two matrices, you have to know their order — how many rows and how many columns each one has. For addition, the order is not a detail: it is the whole condition. Two matrices can be added only when their orders match exactly.
Concept
Reading the order
The order of a matrix is written rows × columns — rows first, always. Count across for rows, down for columns:
This has 2 rows and 3 columns, so its order is 2 × 3. Note that 2 × 3 and 3 × 2 describe genuinely different shapes — the order is not just a pair of numbers you can swap.
A matrix with the same number of rows as columns is called a square matrix. One with a single row is a row matrix; one with a single column is a column matrix.
Concept
When addition is possible
Two matrices can be added only if they have the same order. If the orders differ, the sum is simply undefined — there is no partial answer, no convention for filling gaps.
When the orders match, every entry in the first matrix has exactly one partner in the second — the entry sitting in the same row and column.
The rule itself is then straightforward: add the entries that occupy the same position. The result has the same order as the two matrices you started with.
Example
Adding two matrices
Find , where
Both are 2 × 3, so the sum is defined and will also be 2 × 3.
Top row: %20%3D%201%20%7D)
, 
, 
.
Bottom row: 
, %20%3D%20-1%20%7D)
, 
.
⟹ 
Note
Subtraction and scalar multiplication
Subtraction follows the same position-by-position rule, and needs the same matching orders. Using the matrices above:
Multiplying by a single number (a scalar) has no order restriction at all — multiply every entry:
Example
When the sum does not exist
Can be found, where
is the 2 × 3 matrix above and
The orders are not equal — 2 × 3 is not the same as 3 × 2.
The entry in row 3 of 
has no partner in 
, which has only two rows.
⟹ A + C is undefined
Having the same number of entries is not enough. Both matrices hold six numbers, but they are arranged differently, and the arrangement is what addition depends on.
Example
Combining the two operations
Find , where
is as above and
.
Both are 3 × 2, so the sum will be defined. Do the scalar multiplication first.
Now add position by position: 
, 
, %20%3D%200%20%7D)
, 
, 
, 
.
⟹ 
Note
Properties worth knowing
Order does not matter — 
, because ordinary addition of the paired entries is commutative.
Grouping does not matter — %20%2B%20E%20%3D%20A%20%2B%20(B%20%2B%20E)%20%7D)
.
There is a zero matrix — a matrix of all zeros, of matching order, leaves any matrix unchanged when added.
Every matrix has a negative — flip the sign of every entry and the sum is the zero matrix.
Matrix addition behaves much like ordinary addition. Matrix multiplication, by contrast, does not — there the order of the factors matters a great deal.
Note
Mistakes to avoid
Stating the order as columns × rows — it is rows first.
Assuming 2 × 3 and 3 × 2 can be added because both hold six numbers.
Adding entries that are not in the same position, such as pairing along a diagonal.
Losing a minus sign: , not 5.
Multiplying only the first entry by the scalar instead of every entry.
Summary
- The order of a matrix is rows × columns, in that order; 2 × 3 and 3 × 2 are different shapes.
- Two matrices can be added or subtracted only when their orders are identical.
- Add or subtract entry by entry, pairing entries in the same row and column; the answer keeps the same order.
- Multiplying by a scalar multiplies every entry and works for any order.
- Matrix addition is commutative and associative, has a zero matrix, and every matrix has a negative.