ECON1202 Chap.6 Matrix Algebra and Multiplication
Matrix Algebra and Multiplication
Matrices organise values by rows and columns whose labels carry meaning. Addition requires equal dimensions and operates entry by entry. Multiplication is different: each output entry is a row-column dot product, so the number of columns in the left matrix must equal the number of rows in the right matrix. The unmatched outer dimensions become the result size.
Matrix multiplication is associative and distributive but generally not commutative. Reversing factors can change the question, change the value or make the product undefined. The transpose of a product also reverses order. In applied models, a quantity matrix multiplied by a price, cost or profit vector answers a different economic question while preserving the same row structure.
Units travel through the dot product, and the final vector must be reattached to its row labels. When translating systems into Ax=b, explicit zero coefficients preserve the declared unknown order. A strong solution prints dimensions before multiplying, expands at least one dot product and interprets every output entry with its unit and role.
What this chapter covers
- 01
Matrix dimensions and indexing
- 02
Addition and scalar multiplication
- 03
Product conformability
- 04
Row-column dot products
- 05
Non-commutative order
- 06
Transpose identities
- 07
Applied quantity-vector models
- 08
Systems written as Ax=b
Worked example · free
Turn quantities and prices into row totals
- +1The quantity matrix is 2 by 3 and the price vector is 3 by 1, so the output is 2 by 1.
- +1First store: 4 x 8 + 2 x 3 + 5 x 4 = 58.
- +1Second store: 3 x 8 + 6 x 3 + 1 x 4 = 46.
- +1Interpret the vector [58,46]T as dollars paid by the stores in the original row order.
Key terms
- Matrix order
- The number of rows followed by the number of columns in a matrix.
- Conformable product
- A matrix product whose inner dimensions match.
- Dot product
- The sum of paired products from one row and one column.
- Identity matrix
- A square matrix that leaves a conformable matrix unchanged under multiplication.
- Transpose
- The matrix formed by turning rows into columns and columns into rows.
- Coefficient matrix
- The matrix containing the coefficients of variables in a linear system.
Matrix Algebra and Multiplication FAQ
Why is matrix multiplication not entrywise?
Its purpose is to combine row profiles with column profiles through dot products. Entrywise multiplication answers a different operation and does not produce the transformation or system relationships represented by ordinary matrix multiplication.
How do I know whether AB exists?
Write A as m by n and B as r by p. AB exists only when n equals r, and its size is m by p. The same check must be repeated for BA because reversing the order changes the inner dimensions.
Why does factor order matter in an applied model?
Rows and columns represent different categories. A quantity matrix times a price vector returns one value per quantity row. Reversing the factors changes which categories interact and may no longer represent a meaningful or defined question.
What should an interpreted matrix answer include?
State the output size, preserve the row or column labels, and attach units. Expand one dot product so the source of each term is visible, then translate the resulting vector back into the original category order.
How do I show that a matrix product is meaningful?
Write the dimensions of both factors and name the shared inner category, not just its size. Confirm that the column labels of the left matrix align in the same order with the row labels of the right matrix. State the dimensions and labels of the output. Expand at least one entry as a row-column dot product, preserving signs and units, and place it according to its source row and destination column.
Keep the factor order fixed through algebra; only parentheses may move under associativity. If a transpose is taken, reverse factor order as well as transposing each factor. Interpret the resulting row, column or scalar in the context and use dimensions to check every intermediate expression.
Exam move
Label matrix rows and columns before doing algebra. Practise dimension checks separately from arithmetic by predicting whether AB and BA exist and what sizes they have. For a requested entry, underline only the row and column that contribute, then calculate their dot product without building unused entries. Pair every numeric product with an interpretation problem so units and labels remain attached.
Rehearse the non-commutative identities, especially the reversed order in the transpose of a product and the four-term expansion of a matrix square. When writing Ax=b, declare the unknown order and retain zero coefficients. Verify applied outputs by checking that the result has one entry for the intended category and by recomputing one row manually.
Dimensional validity prevents impossible products, but it does not permit factors to commute, so preserve order throughout.
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