How do matrix dimensions tell me whether two matrices can be added?
Quick answer
Two matrices can be added only when they have the same number of rows and the same number of columns. Compare their dimensions before doing any arithmetic. If they match, add entries in corresponding positions, and keep that same dimension for the result. For example, two 2×2 matrices can be added entry by entry; a 2×3 matrix and a 3×2 matrix cannot.
The question, the details, the next step.
What exactly should I compare when I check whether two matrices can be added? I keep looking at the total number of entries and getting unsure.
Compare the row count and column count separately—not just the total number of entries. A matrix’s dimension is written as rows × columns: for instance, 2×3 means 2 rows and 3 columns. Addition is possible only if both matrices have matching row counts and matching column counts. If they do, add entries in corresponding positions, and the sum has the same dimensions. [S1]
A quick check is to write the dimensions above the matrices before calculating. If either number differs, stop: those matrices cannot be added.
Could two matrices have the same number of entries but still be incompatible? Can you show me a small example?
Yes. Consider a 2×3 matrix and a 3×2 matrix. Each has six entries, but their row and column counts are arranged differently. The dimensions do not match, so they cannot be added. The total entry count alone is not enough; you need both parts of the dimension to agree. [S1]
For contrast, a 2×3 matrix and another 2×3 matrix are compatible. Each position in one has a corresponding position in the other.
My matrices are the same size. How do I do the actual addition, and what size should I give the answer?
Add each entry to the entry in the same row and column of the other matrix. Here is a complete 2×2 example:
A = [[4, 1], [3, 2]] B = [[5, 9], [0, 7]]
Both are 2×2, so they are compatible. Add corresponding entries: 4+5=9, 1+9=10, 3+0=3, and 2+7=9. Therefore:
A + B = [[9, 10], [3, 9]]
The result is also 2×2. Matching dimensions make it possible to pair every entry, and the result retains those dimensions. [S1]
Can I use a simple routine to avoid mistakes on homework, especially when the matrices look different?
Use this three-step routine:
- Count rows and columns in each matrix.
- Write each dimension in rows × columns order and compare both numbers.
- If they match, add corresponding entries and label the result with that same dimension. If they do not match, state that the sum is undefined for that pair rather than trying to combine entries.
For example, if your matrices are 2×3 and 3×3, the row counts differ, so you can stop before calculating. The dimension check decides whether addition is allowed; it does not require you to add any entries first. [S1]
A practical checklist
Count the rows in each matrix.
Count the columns in each matrix.
Write each size as rows × columns and compare both numbers.
If both dimensions match, add entries at corresponding row and column positions.
Label the sum with the same dimensions as the original matrices.
A few more useful answers
If both matrices have six entries, does that mean I can add them?
No. Their row counts and column counts must each match; equal total entry counts are not enough. For example, 2×3 and 3×2 matrices have the same number of entries but cannot be added. [S1]
Does the result get a different dimension after addition?
No. When the matrices are compatible, the sum has the same dimensions as both matrices being added. [S1]
Sources
References used for this guide.
- [S1] OpenStax Algebra and Trigonometry: Matrices and Matrix Operations openstax.org · Checked