A matrix product is a sequence of row-by-column calculations. Before multiplying any entries, check whether the product exists and what shape its answer must have. This short lesson targets the setup errors that survive even when arithmetic is accurate.
Curriculum checked 2 October 2026 · original practice for selected skills.
You will need: Signed arithmetic, brackets and the meaning of a row and a column.
Attempt each task before opening a hint. If you need help, reveal the first hint, then the setup, then compare your written steps with the solution. Finish with the retests without referring back.
A matrix with two rows and three columns has order 2 × 3. The first number always counts rows.
For a product AB, the number of columns of A must equal the number of rows of B. The answer keeps the outer dimensions.
An entry in AB is the sum of products from one row of A and one column of B. It is not entry-by-entry multiplication.
The inner dimensions match: 2 and 2.
The single output entry is .
A 1 × 2 matrix times a 2 × 1 matrix gives a 1 × 1 matrix.
Keep the negative sign attached to the second entry.
Answer: The 1 × 1 matrix [2].
Before moving on: can you explain why your method works, as well as give the answer?
Write a method as well as an answer. The tasks change the reasoning, not just the numbers.
Compare the inner dimensions separately for each order.
AB compares 3 with 3; BA compares 4 with 2.
The inner dimensions agree, so retain 2 and 4 as the output dimensions.
Four columns cannot be paired with two rows.
Answer: AB exists and has order 2 × 4; BA is undefined.
Before moving on: can you explain why your method works, as well as give the answer?
Use each row of the first matrix once.
The top entry is 1 × 4 + 2 × (−1).
Pair matching positions and add.
The zero contributes nothing.
Answer:
Before moving on: can you explain why your method works, as well as give the answer?
Find the top-right entry in each product first.
For AB that entry is 1 × 0 + 1 × 3; for BA it is 2 × 1 + 0 × 1.
One unequal entry is enough to disprove equality.
The top-right entries differ, although both products exist.
Answer: AB ≠ BA.
Before moving on: can you explain why your method works, as well as give the answer?
A tempting claim: “Multiply matching entries to obtain AB.”
That operation ignores the row-column pairing and can give an impossible output shape.
These changed questions test whether you can reconstruct the method. If you use a hint, record where you got stuck and retry later. Completing this small set does not establish full chapter mastery.
Two products contribute to one entry.
Use 3(−2) + 2(5).
Combine a negative and a positive product.
The result is 1 × 1.
Answer: [4]
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
Both inner-dimension checks succeed.
Keep the outer dimensions in their original order.
Three rows remain from C and three columns from D.
Two rows remain from D and two columns from C.
Answer: CD is 3 × 3; DC is 2 × 2.
Could you solve this without the earlier example? If not, revisit the first line where you got stuck.
If a retest exposed the same error, rewrite the first incorrect step and explain its correction aloud. If both were independent, return to a mixed exercise or a missed paper question. A parent can ask what changed in the method rather than only asking for the answer.
Yes. For example, the diagonal matrices diag(1,0) and diag(0,1) are both non-zero, but their product is the zero matrix. Ordinary-number cancellation cannot be assumed for matrices.
Yes. Use the two print buttons above. The question sheet leaves working space and hides solutions; the worked-solutions option includes the solution steps. Your browser can save either view as a PDF.
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