01 / The problem and its objects
Which number are we looking for?
Start with A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]]. These grids of numbers are matrices. Their product is another matrix, C = AB. Find its entry in row 2, column 1 before reading the compact rule for that calculation.
Here a capital letter names a whole matrix. C has one row for each row of A and one column for each column of B: two rows and two columns, with four places to fill.
A lowercase letter with two subscripts names one entry. The first subscript gives its row; the second gives its column. Subscripts may be numbers or letters standing for numbers. We write the chosen entry of C as c21.
Matrix A
| col 1 | col 2 | col 3 | |
|---|---|---|---|
| row 1 | 2 | 1 | 3 |
| row 2 | 4 | 2 | 1 |
Matrix B
| col 1 | col 2 | |
|---|---|---|
| row 1 | 1 | 2 |
| row 2 | 3 | 1 |
| row 3 | 2 | 4 |
Matrix C
| col 1 | col 2 | |
|---|---|---|
| row 1 | ? | ? |
| row 2 | ? | ? |
02 / A labeled diagram
Which numbers meet?
A's second row is (4, 2, 1), across the page. B's first column is (1, 3, 2), down the page. Pair numbers at the same position in those two strips.
- Position 1A row 2: 4B column 1: 1
- Position 2A row 2: 2B column 1: 3
- Position 3A row 2: 1B column 1: 2
03 / Worked arithmetic
Calculate that one entry.
Multiply within each matched pair, then add the three products. Keep the destination at row 2, column 1 throughout.
- Name the address
c21 = the entry in C at row 2, column 1
The first subscript gives the row and the second gives the column. This address asks for one number, not the whole matrix C.
- Choose the strips
A row 2 = (4, 2, 1); B column 1 = (1, 3, 2)
The first number in each strip is a pair, then the second numbers, then the third numbers.
- Multiply matching positions
4 × 1, 2 × 3, 1 × 2
Position 1 meets position 1, and so on. The three pairs give three products, rather than every possible combination.
- Add the products
c21 = 4 × 1 + 2 × 3 + 1 × 2 = 4 + 6 + 2 = 12
This gives one entry, not the whole matrix C.
One entry foundc21 = 12
04 / The compact instruction
How does the formula record those steps?
The arithmetic above found one entry. For these matrices, this line describes the same move at any chosen row and column of C:
Read aloud
To find the entry of C in row i and column j, let z run through 1, 2, and 3. At each position z, multiply the entry in row i of A by the matching entry in column j of B. Add those three products.
The left side, cij, names one answer. The right side says how to make it. The letters i and j stay fixed for that answer; z moves through matching positions.
- A, B, C
- Whole matrices: grids of numbers. Here C is the matrix product AB.
- aiz, bzj, cij
- Individual entries. The first subscript gives a row and the second gives a column. For example, c_21 is one number in row 2, column 1 of C.
- i, j
- The fixed address of the one output entry we are finding: row i and column j.
- z
- The running index. It changes from 1 to 3 and links corresponding positions in A's row and B's column.
- Σ
- A request to add. The lower and upper labels set the first and last values of the running index. The sum covers the complete product a_iz b_zj.
- =
- Equality: the expression on the right gives the value of the entry on the left. In a definition, some authors use := to flag that they are introducing a name.
01 / A reading lens
Hold the destination still. Move through the matching positions.
To find the entry of C in row i and column j, let z run through 1, 2, and 3. At each position z, multiply the entry in row i of A by the matching entry in column j of B. Add those three products.
Fixed address · For c_21, i = 2 and j = 1 throughout.
Moving position · Only z changes: 1 → 2 → 3.
a21 × b11 4 × 1 = 4
Conventions need context
Below or above. A subscript often identifies an entry, as in a_21; an exponent often means a power, as in x² = x × x. These are conventions, not a universal visual code. A superscript can also be a component label in another subject, so check the definitions nearby.
Capital or lowercase. Here A names a whole matrix and a_21 names one of its entries. That is a common convention, but an author can choose different letters. Read the local definition before relying on case alone.
Equal or defined as. An equals sign says the two sides have the same value. A colon-equals sign (:=) sometimes announces a definition. In context, an ordinary equals sign may also introduce a definition. Neither sign means “perform these steps in this order” by itself.
05 / The same reading habit
A sum without a matrix.
The running-index idea does not depend on matrices. Read this smaller expression the same way:
Let k take the values 1, 2, 3, and 4. Double each value, then add the four results.
2(1) + 2(2) + 2(3) + 2(4) = 2 + 4 + 6 + 8 = 20
The letter k is a placeholder for the changing value. Calling it t instead would leave the answer at 20. The 2k means 2 times k, not k squared.
06 / A tempting wrong turn
What if every letter moves?
07 / Independent practice
Try the instruction yourself.
The questions move from recognizing an address to applying and transferring the rule. Open a hint only if you need it; each answer includes its reasoning.
In a grid P, which address does p42 name?
Show a hint
Read the first subscript as the row and the second as the column.
Show worked answer
Row 4, column 2 of P.
- P names the whole grid; lowercase p with subscripts names one entry.
- The first subscript 4 chooses the row. The second subscript 2 chooses the column.
Let A be 2 × 3, B be 3 × 2, and C = AB. Its entry rule is cij = 3Σz=1 aiz bzj. While finding c21, which symbols stay fixed and which one runs?
Show a hint
Start with the requested output address 21. Then look at the letter below Σ.
Show worked answer
i = 2 and j = 1 stay fixed; z runs through 1, 2, and 3.
- The destination c_21 fixes row i at 2 and column j at 1.
- The bounds under and over Σ tell z to take the values 1, 2, and 3.
Expand and evaluate 4Σk=1 (k + 1).
Show a hint
Substitute k = 1, then 2, then 3, then 4 into the entire expression (k + 1).
Show worked answer
(1 + 1) + (2 + 1) + (3 + 1) + (4 + 1) = 2 + 3 + 4 + 5 = 14.
- The bounds include both endpoints, so there are four terms.
- Evaluate (k + 1) at each value and add the four results.
With A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]], expand and calculate c12 in C = AB.
Show a hint
The address 12 selects row 1 of A and column 2 of B. Pair their first, second, and third positions.
Show worked answer
c12 = 2 × 2 + 1 × 1 + 3 × 4 = 4 + 1 + 12 = 17.
- For c_12, fix i = 1 and j = 2; z still runs through 1, 2, and 3.
- A's first row is (2, 1, 3). B's second column is (2, 1, 4).
- Multiply corresponding positions and add: 4 + 1 + 12 = 17.
Expand and evaluate 4Σr=2 3r. Does changing the running letter from r to t change the value?
Show a hint
The first value is 2, not 1. Visit r = 2, 3, and 4.
Show worked answer
3(2) + 3(3) + 3(4) = 6 + 9 + 12 = 27. Renaming r to t does not change the value.
- The lower bound starts the sum at 2; the upper bound ends it at 4, giving three terms.
- The running letter is a placeholder. The same three values and rule give the same result with another letter.
A rule says dij = 5Σq=1 xiq yqj. How many products make one entry dij, and does the 5 say how many entries D has?
Show a hint
Count the values of q while holding one output address ij fixed.
Show worked answer
Five products make each chosen entry. The 5 does not say how many entries D has.
- The running index q takes 1, 2, 3, 4, and 5, so the sum has five product terms.
- The size of D comes from its available row and column addresses, which this sum bound alone does not specify.
FIELD NOTES / RECALL
Quick recall · six cards
Read the front, decide on an answer, then reveal the reasoning.
Card 1 of 6
In a grid P, which address does p42 name?
Back / answer
Row 4, column 2 of P.
P names the whole grid; lowercase p with subscripts names one entry. The first subscript 4 chooses the row. The second subscript 2 chooses the column.
Card 1 of 6: In a grid P, which address does p_42 name?. Answer hidden.
Card 1 / recognize
Front / question
In a grid P, which address does p_42 name?
Back / answer
Row 4, column 2 of P.
P names the whole grid; lowercase p with subscripts names one entry. The first subscript 4 chooses the row. The second subscript 2 chooses the column.
Card 2 / recognize
Front / question
Let A be 2 × 3, B be 3 × 2, and C = AB. Its entry rule is c_ij = Σ_(z=1)^3 a_iz b_zj. While finding c_21, which symbols stay fixed and which one runs?
Back / answer
i = 2 and j = 1 stay fixed; z runs through 1, 2, and 3.
The destination c_21 fixes row i at 2 and column j at 1. The bounds under and over Σ tell z to take the values 1, 2, and 3.
Card 3 / apply
Front / question
Expand and evaluate Σ_(k=1)^4 (k + 1).
Back / answer
(1 + 1) + (2 + 1) + (3 + 1) + (4 + 1) = 2 + 3 + 4 + 5 = 14.
The bounds include both endpoints, so there are four terms. Evaluate (k + 1) at each value and add the four results.
Card 4 / apply
Front / question
With A = [[2, 1, 3], [4, 2, 1]] and B = [[1, 2], [3, 1], [2, 4]], expand and calculate c_12 in C = AB.
Back / answer
c_12 = 2 × 2 + 1 × 1 + 3 × 4 = 4 + 1 + 12 = 17.
For c_12, fix i = 1 and j = 2; z still runs through 1, 2, and 3. A's first row is (2, 1, 3). B's second column is (2, 1, 4). Multiply corresponding positions and add: 4 + 1 + 12 = 17.
Card 5 / transfer
Front / question
Expand and evaluate Σ_(r=2)^4 3r. Does changing the running letter from r to t change the value?
Back / answer
3(2) + 3(3) + 3(4) = 6 + 9 + 12 = 27. Renaming r to t does not change the value.
The lower bound starts the sum at 2; the upper bound ends it at 4, giving three terms. The running letter is a placeholder. The same three values and rule give the same result with another letter.
Card 6 / transfer
Front / question
A rule says d_ij = Σ_(q=1)^5 x_iq y_qj. How many products make one entry d_ij, and does the 5 say how many entries D has?
Back / answer
Five products make each chosen entry. The 5 does not say how many entries D has.
The running index q takes 1, 2, 3, 4, and 5, so the sum has five product terms. The size of D comes from its available row and column addresses, which this sum bound alone does not specify.
08 / Keep this page
One-minute reference.
- Name the output first: c_ij is one entry, while C is the whole matrix.
- Subscript ij gives the output address: row i, column j.
- In one sum, i and j stay fixed; z takes every integer from 1 through 3.
- At each z, multiply a_iz by b_zj. Add the complete products.
- Check whether a raised mark is a power or a label from its local definition.
Where it appears again
In the next note, Rows, Columns, and Why You Don't Multiply Everything, this reading rule becomes a complete matrix product. Later, the same habit—identify the object, hold some indices fixed, and let another index vary—helps with matrix transformations and repeated application.
Continue to the matrix multiplication noteFor the standard entry notation and row-column rule, see Interactive Linear Algebra, §3.4 Matrix Multiplication — Georgia Tech. This note’s wording, diagram, and exercises were written for this series.