01 / The objects and the question
What are we multiplying?
The two grids below are matrices A and B. Our task is to multiply A by B and find every number in the result, C = AB. The question marks show what we still need to calculate.
| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 2 | 1 | 3 |
| row 2 | 4 | 2 | 1 |
| col 1 | col 2 | |
|---|---|---|
| row 1 | 1 | 2 |
| row 2 | 3 | 1 |
| row 3 | 2 | 4 |
| col 1 | col 2 | |
|---|---|---|
| row 1 | ? | ? |
| row 2 | ? | ? |
Each letter names a whole matrix. One number inside a matrix is an entry. We name an entry with a lowercase letter and two subscripts: the first gives its row, the second its column. For example, a21 is row 2, column 1 of A, so a21 = 4.
Let us choose C's lower-left question mark as our first entry. It sits in row 2, column 1, so its address is c21. There is nothing special about starting here; the same move will fill the other three cells.
02 / The visual
How does c21 get its value?
The address selects row 2 of A and column 1 of B. Pair first with first, second with second, third with third. Multiply each pair, then add the three products.
| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 2 | 1 | 3 |
| row 2 | 4 | 2 | 1 |
| col 1 | col 2 | |
|---|---|---|
| row 1 | 1 | 2 |
| row 2 | 3 | 1 |
| row 3 | 2 | 4 |
| col 1 | col 2 | |
|---|---|---|
| row 1 | ? | ? |
| row 2 | ? | ? |
Expanded arithmeticc21 = 4 × 1 + 2 × 3 + 1 × 2 = 4 + 6 + 2 = 12
| B pos 1 1 | B pos 2 3 | B pos 3 2 | |
|---|---|---|---|
| A pos 1 4 | 4 × 1 match | 4 × 3 | 4 × 2 |
| A pos 2 2 | 2 × 1 | 2 × 3 match | 2 × 2 |
| A pos 3 1 | 1 × 1 | 1 × 3 | 1 × 2 match |
03 / The worked example
What about the other addresses?
Keep the matrices fixed. Change the row of A or the column of B when the address changes. Each line below makes exactly one entry of C.
Put the four answers at their addresses:C = 11171214
04 / The compact rule
How do we write the same move generally?
Once the row and column operation is clear, one line can stand for it. Suppose A has m rows and n columns, and B has n rows and p columns. Then C has m rows and p columns.
One entry, at address (i, j)
cij = n∑z = 1 aizbzj
Hold i (the row) and j (the column) fixed. Let z move through the matching positions, from 1 to the shared dimension n. Multiply at each position and add.
In our example n = 3. The formula describes one number cij. Calculating it for every allowed address and placing those numbers in their positions assembles the whole matrix C.
05 / Optional step-through
Follow another address
Select an entry of C. The marked row and column will move with it. The explanation above is complete on its own; this control simply lets you retrace any of the four calculations.
Choose an address in C The first subscript selects A's row; the second selects B's column.
Showing c21| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 2 | 1 | 3 |
| row 2 | 4 | 2 | 1 |
| col 1 | col 2 | |
|---|---|---|
| row 1 | 1 | 2 |
| row 2 | 3 | 1 |
| row 3 | 2 | 4 |
Result C select one address
| col 1 | col 2 | |
|---|---|---|
| row 1 | ||
| row 2 |
Address c21 step 1 of 4
Take row 2 of A and column 1 of B. Which positions pair up?
Read the address as row first, column second. For c21, take the second row of A and the first column of B. Keep their positions aligned.
06 / Practice
Try the addresses yourself
Begin with the guided address checks, then work the independent questions on paper. Open each answer to check the method, not only the final number.
Guided / recognize
In A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]], find a_23 and b_32. Say each address as row, column before reading its value.
Show a hint
For a_23, use A's second row and third column. For b_32, use B's third row and second column.
Show the full reasoning and answer
a_23 = 1 and b_32 = 4.
A's second row is (4, 2, 1); its third entry is 1.
B's third row is (2, 4); its second entry is 4. An address names one entry, not the whole matrix.
Guided / recognize
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. For c_12, which row of A and column of B must meet? How many paired products make that one entry?
Show a hint
Read 12 as row 1, column 2. The shared dimension is 3.
Show the full reasoning and answer
Use A row 1, (2, 1, 3), with B column 2, (2, 1, 4). That entry has three paired products.
The first subscript 1 selects A row 1; the second subscript 2 selects B column 2.
Their positions match three times: 2 with 2, 1 with 1, and 3 with 4. The shape's shared 3 sets the count.
Independent / apply
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. Calculate c_22. Write the chosen row and column before the arithmetic.
Show a hint
Take A row 2, (4, 2, 1), and B column 2, (2, 1, 4).
Show the full reasoning and answer
c_22 = 4×2 + 2×1 + 1×4 = 8 + 2 + 4 = 14.
Address 22 fixes row 2 of A and column 2 of B.
The corresponding pairs are 4 with 2, 2 with 1, and 1 with 4. Their products add to 14.
Place 14 at row 2, column 2 of C.
Independent / apply
Let A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]]. Is BA defined? If so, give its shape and top-left entry.
Show a hint
B is 3 × 2 and A is 2 × 3. For the top-left entry, use B row 1 and A column 1.
Show the full reasoning and answer
BA has shape 3 × 3. Its top-left entry is 1×2 + 2×4 = 10.
The adjacent dimensions in (3 × 2)(2 × 3) match at 2; the outside dimensions are 3 and 3.
B row 1 is (1, 2), and A column 1 is (2, 4). Pair their positions and add 2 + 8 = 10.
Independent / apply
Let P = [[1, 0], [2, 3]] and Q = [[4], [5]]. Find PQ and its shape.
Show a hint
Q has one column, so PQ has one column. Make its two entries using each row of P with that column.
Show the full reasoning and answer
PQ = [[4], [23]], a 2 × 1 matrix.
The shapes (2 × 2)(2 × 1) fit, and the outside dimensions give 2 × 1.
The first entry is 1×4 + 0×5 = 4. The second is 2×4 + 3×5 = 8 + 15 = 23.
Put the two entries in one column: [[4], [23]].
Independent / transfer
A is 2 × 3, B is 3 × 2, and D is any 2 × 3 matrix. Which of AB, BA, AD, and DA are defined? Give each valid output shape.
Show a hint
For each order, compare only the adjacent dimensions. Then read the outside dimensions if they match.
Show the full reasoning and answer
AB is defined and is 2 × 2. BA is defined and is 3 × 3. AD and DA are undefined.
AB has shapes (2 × 3)(3 × 2): the adjacent 3s match, giving 2 × 2.
BA has shapes (3 × 2)(2 × 3): the adjacent 2s match, giving 3 × 3.
AD and DA both have shapes (2 × 3)(2 × 3). Their adjacent dimensions are 3 and 2, so neither product is defined.
Independent / transfer
Without calculating entries, describe (4 × 9)(9 × 6): output shape, total output entries, and paired products per entry.
Show a hint
The outside numbers make the output shape. Count its cells, then use the shared dimension for the number of products inside each cell.
Show the full reasoning and answer
The output is 4 × 6, with 24 entries and 9 paired products in each entry.
The adjacent 9s match, so the product is defined. The outside dimensions 4 and 6 give its shape.
A 4 × 6 matrix has 4×6 = 24 entries. The shared 9 means each entry adds nine corresponding products.
Independent / transfer
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. Someone tries nine every-to-every products for c_21. Which row and column should they choose, how should they pair positions instead, and what is c_21?
Show a hint
Write the two three-number strips side by side. Pair only entries at the same position.
Show the full reasoning and answer
Make three corresponding products: 4×1 + 2×3 + 1×2 = 12.
For c_21, A row 2 is (4, 2, 1) and B column 1 is (1, 3, 2).
Pair first with first, second with second, and third with third, giving 4, 6, and 2.
Add those three products to get 12. The shared dimension is 3, so nine every-to-every products answer a different question.
FIELD NOTES / RECALL
Practice cards from note 002
Read the front, decide on an answer, then reveal the reasoning.
Card 1 of 8
In A = 213421 and B = 123124, find a23 and b32. Say each address as row, column before reading its value.
Back / answer
a23 = 1 and b32 = 4.
A's second row is (4, 2, 1); its third entry is 1. B's third row is (2, 4); its second entry is 4. An address names one entry, not the whole matrix.
Card 1 of 8: In A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]], find a_23 and b_32. Say each address as row, column before reading its value.. Answer hidden.
Card 1 / recognize
Front / question
In A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]], find a_23 and b_32. Say each address as row, column before reading its value.
Back / answer
a_23 = 1 and b_32 = 4.
A's second row is (4, 2, 1); its third entry is 1. B's third row is (2, 4); its second entry is 4. An address names one entry, not the whole matrix.
Card 2 / recognize
Front / question
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. For c_12, which row of A and column of B must meet? How many paired products make that one entry?
Back / answer
Use A row 1, (2, 1, 3), with B column 2, (2, 1, 4). That entry has three paired products.
The first subscript 1 selects A row 1; the second subscript 2 selects B column 2. Their positions match three times: 2 with 2, 1 with 1, and 3 with 4. The shape's shared 3 sets the count.
Card 3 / apply
Front / question
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. Calculate c_22. Write the chosen row and column before the arithmetic.
Back / answer
c_22 = 4×2 + 2×1 + 1×4 = 8 + 2 + 4 = 14.
Address 22 fixes row 2 of A and column 2 of B. The corresponding pairs are 4 with 2, 2 with 1, and 1 with 4. Their products add to 14. Place 14 at row 2, column 2 of C.
Card 4 / apply
Front / question
Let A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]]. Is BA defined? If so, give its shape and top-left entry.
Back / answer
BA has shape 3 × 3. Its top-left entry is 1×2 + 2×4 = 10.
The adjacent dimensions in (3 × 2)(2 × 3) match at 2; the outside dimensions are 3 and 3. B row 1 is (1, 2), and A column 1 is (2, 4). Pair their positions and add 2 + 8 = 10.
Card 5 / apply
Front / question
Let P = [[1, 0], [2, 3]] and Q = [[4], [5]]. Find PQ and its shape.
Back / answer
PQ = [[4], [23]], a 2 × 1 matrix.
The shapes (2 × 2)(2 × 1) fit, and the outside dimensions give 2 × 1. The first entry is 1×4 + 0×5 = 4. The second is 2×4 + 3×5 = 8 + 15 = 23. Put the two entries in one column: [[4], [23]].
Card 6 / transfer
Front / question
A is 2 × 3, B is 3 × 2, and D is any 2 × 3 matrix. Which of AB, BA, AD, and DA are defined? Give each valid output shape.
Back / answer
AB is defined and is 2 × 2. BA is defined and is 3 × 3. AD and DA are undefined.
AB has shapes (2 × 3)(3 × 2): the adjacent 3s match, giving 2 × 2. BA has shapes (3 × 2)(2 × 3): the adjacent 2s match, giving 3 × 3. AD and DA both have shapes (2 × 3)(2 × 3). Their adjacent dimensions are 3 and 2, so neither product is defined.
Card 7 / transfer
Front / question
Without calculating entries, describe (4 × 9)(9 × 6): output shape, total output entries, and paired products per entry.
Back / answer
The output is 4 × 6, with 24 entries and 9 paired products in each entry.
The adjacent 9s match, so the product is defined. The outside dimensions 4 and 6 give its shape. A 4 × 6 matrix has 4×6 = 24 entries. The shared 9 means each entry adds nine corresponding products.
Card 8 / transfer
Front / question
Let A = [[2, 1, 3], [4, 2, 1]], B = [[1, 2], [3, 1], [2, 4]], and C = AB. Someone tries nine every-to-every products for c_21. Which row and column should they choose, how should they pair positions instead, and what is c_21?
Back / answer
Make three corresponding products: 4×1 + 2×3 + 1×2 = 12.
For c_21, A row 2 is (4, 2, 1) and B column 1 is (1, 3, 2). Pair first with first, second with second, and third with third, giving 4, 6, and 2. Add those three products to get 12. The shared dimension is 3, so nine every-to-every products answer a different question.
Where it appears again
This row-and-column move is a foundation
A matrix product can describe the composition of transformations. In a later note, a square matrix A and nonzero eigenvector v satisfy Av = λv for some scalar λ. Step zero leaves v unchanged; for whole-number steps n ≥ 1, repeated application gives Aⁿv = λⁿv. This is repeated matrix application, not entrywise squaring. A real matrix need not have a real eigenvector.
Continue through the six-note volume
Each guide begins with a plain-language question and develops one worked example before introducing compact notation.
See the complete reading order and all six guides →
For real scaling: λ = 2 doubles the size; ½ halves it; −½ halves it and reverses direction each time; 1 leaves v unchanged; −1 reverses direction without changing size; and 0 sends v to the zero vector, which has no direction. With λ = −½ the first four states are v, −v/2, v/4, −v/8.
One-minute reference
The move in four lines
- Check shape: the adjacent dimensions must match. (2 × 3)(3 × 2) gives a 2 × 2 output.
- Choose one address c_ij: i fixes the row of A, and j fixes the column of B.
- Pair position 1 with 1, position 2 with 2, and so on through the shared dimension. Multiply within each pair and add.
- One sum gives one c_ij. Repeat over all valid output addresses to assemble C.