Matrix multiplication homework: keep the order

Matrix multiplication homework starts with dimensions

The size m × n means m rows followed by n columns. In an entry a_ij, i identifies the row and j the column. Take A = [1, 2, −1; 0, 3, 2], where a semicolon separates rows. It has two rows and three columns, so its size is 2 × 3. Its entry a_23 is 2, whereas a_13 is −1. The subscripts name positions; they are not factors multiplying the letter a.

OpenStax’s explanation of matrix products gives the compatibility rule and row-by-column definition. If A has size m × r and B has size r × n, AB exists and has size m × n. Matching inner dimensions permit pairing the entries; the outer dimensions describe the output. This is a definition of the ordinary matrix product. Do not borrow the equal-size condition for addition, which is a different operation with different requirements.

Decide the output shape before filling it

Let B = [2, 1; −1, 4; 3, 0]. Its size is 3 × 2. In (2 × 3)(3 × 2), the inner threes match, so AB is a 2 × 2 matrix. Draw four empty output positions on paper. Each output row belongs to a row of A; each output column belongs to a column of B. This simple preparation catches an incorrectly copied matrix before several lines of unnecessary calculation.

Two matrices of the same rectangular size need not be compatible. A 2 × 3 matrix cannot multiply another 2 × 3 matrix in that order because 3 does not equal 2. Conversely, differently sized matrices can be compatible, as our A and B show. Count positions even when they contain zero. Removing a zero entry changes the rectangle and destroys the positional correspondence. A missing term is not permission to slide every following entry left.

Build one entry from a complete row and column

Call the product C = AB. For c_11, use row 1 of A, (1, 2, −1), and column 1 of B, (2, −1, 3). Multiply corresponding terms of those two lists and add: c_11 = 1 × 2 + 2 × (−1) + (−1) × 3 = 2 − 2 − 3 = −3. There are three scalar products because the matching inner dimension is three, although C itself has only two columns.

For c_12, keep the same row of A and move to column 2 of B, (1, 4, 0). Thus c_12 = 1 × 1 + 2 × 4 + (−1) × 0 = 1 + 8 + 0 = 9. Write the selected row and column separately if your eye keeps jumping to B’s first row. An output address tells you which two lists to combine, not which single pair of original entries to multiply.

Finish the rectangle and locate sign mistakes

For the second output row, use row 2 of A, (0, 3, 2). Column 1 gives c_21 = 0 × 2 + 3 × (−1) + 2 × 3 = 3. Column 2 gives c_22 = 0 × 1 + 3 × 4 + 2 × 0 = 12. Therefore AB = [−3, 9; 3, 12]. Keep each result in its original row-column address; moving the four correct numbers to different positions still changes the matrix.

Check a suspicious entry by expanding its scalar products before adding them. For example, the middle contribution to c_21 is −3, not +3, while its final contribution is +6. If you obtained 9, the first question is where that negative sign went. If all four entries look wrong, revisit the operation and orientation before blaming arithmetic. Multiplying matching positions, adding the matrices, and multiplying a matrix by one number follow different rules.

Reversing the factors changes the task

In BA, the sizes are (3 × 2)(2 × 3), so the product exists but has size 3 × 3. Its first row is (2, 1) multiplied by each column of A, giving (2, 7, 0). Repeating for the other rows produces BA = [2, 7, 0; −1, 10, 9; 3, 6, −3]. This cannot equal our 2 × 2 product AB. Both calculations are legitimate, but they answer different ordered questions.

Sometimes only one order exists. A 2 × 3 matrix may multiply a 3 × 1 column to give a 2 × 1 column. Reversing those factors would require the inner dimensions 1 and 2 to match, which they do not. The undefined reverse product is not a matrix of zeros. Do not silently transpose a factor to make a requested product possible; that would replace the given operation with a different one.

Equal sizes do not guarantee equal reversed products

Even two square factors can change value when reversed. Set X = [1, 1; 0, 1] and Y = [2, 0; 0, 3]. Both orders have size 2 × 2, but XY = [2, 3; 0, 3] whereas YX = [2, 2; 0, 3]. The upper-right entries differ. A dimension check is therefore necessary for a valid product but cannot certify its arithmetic or prove that factors may be exchanged.

Noncommutative means there is no general law AB = BA. It does not mean every pair of square matrices produces different reversed products. The identity matrix I = [1, 0; 0, 1] satisfies IX = XI = X for every 2 × 2 matrix X. Preserve the original factor order even when a special example happens to commute. An exception illustrates the distinction; it does not establish an exchange rule for the next exercise.

Attach units to an applied matrix product

Suppose two fictional orders contain quantities of three products: Q = [2, 1, 0; 1, 0, 3]. Let the matching unit prices be P = [4; 5; 2] in currency units per item. QP has size 2 × 1 and gives order totals [13; 10]. The first is 2 × 4 + 1 × 5 + 0 × 2; the second is 1 × 4 + 0 × 5 + 3 × 2. These are invented arithmetic amounts, not actual product prices.

The categories must appear in the same order across Q’s columns and P’s rows. Dimensions alone cannot detect swapping the second and third product labels while leaving the numbers fixed. Each quantity must pair with its own unit price, and every term added into an order total must have the same currency unit. If a number is a percentage, length or different currency, resolve its meaning before inserting it. A structurally valid multiplication can still represent the wrong model.

Capture the whole task before asking Eqora

Begin with your own attempted product and the first doubtful entry. If your course permits AI support, photograph one complete exercise with both matrices, their brackets and the requested order visible. Avoid personal details, glare and faint minus signs. Preserve row breaks and zeros. Before reading any suggested steps in Eqora, compare the captured task with the original: A must still have two rows, B three, and the question must still request AB rather than BA.

Then ask a focused question such as: “For c_12, which row and column are selected, and what are the three scalar products before adding?” This is a proposed learning prompt, not a quoted app response. Check any explanation against (1, 2, −1) and (1, 4, 0). Our guide to asking follow-up math questions shows how naming the exact obstacle can make an explanation easier to assess than requesting another complete answer without identifying the uncertain step.

If the input was misread, correct or recapture it before trusting the result. If a response ignores a negative sign, multiplies matching positions, or reverses the factors, stop and compare it with the definition or your teacher’s method. Paper suffices to solve these examples without a paid service; that does not claim unlimited app access is free. Explain the corrected entry in your own words, then close the assisted solution before doing another problem.

Solve a changed product without the worked answer

Try M = [2, −1; 1, 3] multiplied by N = [4; −2]. Predict a 2 × 1 result first. The top entry is 2 × 4 + (−1) × (−2) = 10, and the bottom entry is 1 × 4 + 3 × (−2) = −2. Hence MN = [10; −2]. Notice the different roles of the two negative signs in the top entry and the single negative contribution in the bottom entry.

Check independently using columns: four copies of M’s first column minus two copies of its second column give 4(2, 1) − 2(−1, 3) = (10, −2). This reorganization checks the same product through a different grouping of scalar arithmetic. Our guide to checking answers without a key explains why repeating an identical line is weaker than a second representation. Finish by describing the permitted dimensions, selected row-column pairs and unchanged order, rather than memorizing the final two numbers.