Matrix multiplication describes successive linear operations. For A=[[1,2],[0,3]], applying A to (4,1) gives (6,3): each output is the dot product of a row with the input. Throughout this unit, nested brackets list rows; vectors are interpreted as columns even when printed as coordinate tuples.
Predict
Predict the shape before computing. If A is m×n, it accepts n input coordinates and returns m. For B of size n×p, AB is m×p. The entry (AB)ᵢⱼ sums AᵢₖBₖⱼ over k=1,…,n. Matrix addition instead requires identical shapes and adds corresponding entries.
Experiment
Take B=[[2,0],[1,1]] and x=(1,2). Compute Bx=(2,3), then A(Bx)=(8,9). Independently form AB=[[4,2],[3,3]] and multiply it by x. The agreement checks composition order. Try BA and explain why it represents a different sequence.
Understand
Worked reasoning: the jth column of B is the output B eⱼ on the jth standard basis vector. Applying A to that column gives A(B eⱼ), precisely the jth column of AB. This column view often makes multiplication more understandable than a grid of isolated sums.
Matrix multiplication distributes over addition and is associative for compatible shapes: A(Bx)=(AB)x. It is not generally commutative. Even when AB exists, BA may not; when both exist they can differ. The transpose Aᵀ interchanges rows and columns and satisfies (AB)ᵀ=BᵀAᵀ.
Invalid shortcut: multiply entries in matching positions to compose maps. That defines a different operation and ignores the sums over intermediate coordinates.
Apply
Write input, intermediate and output dimensions beside each practice computation. Verify one product column by applying the maps to a standard basis vector. Keep a minus sign attached to its entry rather than to an entire row unless a row operation says so.
Continue
Depth: diagonal matrices rescale coordinate directions; permutation matrices reorder them. Products of elementary matrices represent the row operations used in elimination. These links let you move between computational rules and concrete transformations.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
A=[[1,2],[0,3]] and B=[[2,0],[1,4]]. Enter the (1,2) entry of AB.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
For A=[[2,−1],[3,0]] and x=(4,5), enter the second coordinate of Ax.
Integers, fractions and finite decimals are accepted. Expressions such as sqrt(2) are not parsed by this checker.
A is 3×2 and B is 2×4. What is the shape of AB?
If x first passes through B and then A, the composite output is:
Use A=[[1,1],[0,1]] and B=[[1,0],[0,2]] to show AB≠BA.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove the jth column of AB is A times the jth column of B, for compatible sizes.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
