Which number laws survive for matrices?

Try

Two workshops assemble two kinds of kits. Matrix A=[[2,1],[1,3]] records kit counts, with rows workshops and columns kit types. Matrix B=[[3,2],[1,4]] records parts per kit, with rows those same kit types and columns screws, clips. How many parts does each workshop need?

For the first workshop’s screws, multiply and add: 2·3+1·1=7. Doing this for every workshop and part gives AB=[[7,8],[6,14]]. The inner labels match: A’s kit columns align with B’s kit rows. The output rows are workshops and output columns are parts.

Predict

Predict whether BA describes the same thing. Even if both arrays are 2×2 and the reversed product is algebraically defined, the reversed labels do not express the same contextual calculation. In general, order changes the matrix product.

Predict which matrix preserves every column vector when multiplied by it. The zero matrix cannot: it sends every vector to zero. The identity matrix I=[[1,0],[0,1]] selects and preserves the original coordinates.

Experiment

Use a row/column ledger to compute every entry of the fixed product.

Output entry	Calculation	Value
Workshop 1, screws	2·3+1·1	7
Workshop 1, clips	2·2+1·4	8
Workshop 2, screws	1·3+3·1	6
Workshop 2, clips	1·2+3·4	14

Now compute BA as bare number arrays: [[8,9],[6,13]]. It differs from AB exactly, with no rounding involved. Then calculate AI and IA separately. Contrast both with A+0 and A0. The additive identity is the zero matrix of matching shape, while the multiplicative identity is the appropriately sized square identity matrix.

Understand

If A is m×n and B is n×p, then AB is m×p. Entry (i,j) is the sum, over k from 1 to n, of A(i,k)B(k,j). The matching inner dimension counts terms in the row/column pairing. For addition, both dimensions must match instead.

Distributivity follows from real arithmetic entry by entry. For B,C both n×p, each term A(i,k)(B(k,j)+C(k,j)) distributes, giving A(B+C)=AB+AC. If D,E are m×n and F is n×p, distribute the first factor instead to get (D+E)F=DF+EF. For associativity, let C additionally have size p×q. Both ((AB)C)(i,j) and (A(BC))(i,j) sum the same products A(i,k)B(k,l)C(l,j) over k and l; only the order of the finite sums changes. Thus regrouping is valid. It does not authorize reordering A,B,C.

For a square matrix [[a,b],[c,d]], define determinant Δ=ad−bc. If Δ≠0, multiplication on both sides verifies the inverse (1/Δ)[[d,−b],[−c,a]]. If Δ=0 and the first row is nonzero, the nonzero column (−b,a) is sent to (0,Δ)=(0,0). If the first row is zero but the second is nonzero, the column (−d,c) is sent to zero. If the whole matrix is zero, use (1,0). In each singular case an inverse would send the equality Av=0 back to v=0, a contradiction. This proves the determinant criterion for 2×2 matrices; higher sizes require further definitions.

A nonzero matrix can therefore fail to be invertible. For A=diag(1,0) and B=diag(0,1), AB=0 even though neither factor is zero. Cancellation laws for nonzero real numbers cannot be transferred to all nonzero matrices.

Apply

State shapes before multiplying and check both products when proposing a 2×2 inverse. Keep an exact noncommuting example beside a general distributivity argument: a counterexample refutes a universal claim, while one matching numerical example cannot prove a universal law.

For a data product, also identify the summed-over label and the two surviving labels. A row-by-column answer without those meanings may be algebraically correct but contextually inappropriate.

Continue

This entry-level bridge develops N-VM.C.8–10. Continue to matrix products and composition and determinants and volume for further transformation and area reasoning; these direct links do not require completing the whole linear algebra course first.

The mixed assessment asks for fresh constructions and proofs across this unit. Its public worked guidance supports practice. Completion and saved answers do not by themselves establish independently reviewed mathematical understanding.