An ordered rectangular array, and what its order tells you before anything else

What to assume they know

Reading a two-way table with labelled rows and columns
Ordered pairs, and the fact that swapping the two entries makes a different pair
Factorising a small whole number into a product of two factors
Absolute value of a real number, from Class XI
Subscript notation, and a rule that assigns a value to each index
Cartesian coordinates of a point in the plane
Surds, decimals and simple algebraic and trigonometric expressions as things that can sit where a number sits

What they should be able to do

Explain what an arrangement records that a plain list of the same numbers does not
State Definition 1 and say what work the word ordered is doing in it
Name the horizontal and vertical lines of a printed array correctly
Read the order of a printed matrix, rows before columns, and write it as m by n
Locate the entry a sub i j and say which line each index selects
Write the general array with double subscripts, and read off an arbitrary row and an arbitrary column from it
Count the entries of a matrix from its order, and run that count backwards to list every order a given number of entries permits
Construct a matrix from a rule that computes each entry from its two indices
Represent a point and the vertices of a plane figure as matrices, in both of the layouts the chapter offers
State the two working restrictions the chapter imposes on itself, and notice the printed example that sits outside one of them

Where it usually goes wrong

"Order means columns by rows, or it does not matter which." It is rows first, always, and it matters constantly: a three by two matrix and a two by three matrix cannot be added to each other, and the chapter's own two arrangements of the notebook data are exactly this pair.
"The two notebook arrangements are the same matrix, because the data is the same." They are two matrices of different orders, and the chapter prints both precisely so that the difference is visible before any operation demands it.
"A matrix is a way of writing a number, like a determinant." Nothing in this chapter evaluates a matrix to a single number, and the word determinant does not occur anywhere in Part I pp. 34–75 — checked against the extracted text of all forty-two pages and against the page image of every one of them. A matrix is the arrangement itself.
"Eight entries give two orders, one by eight and four by two." Four. The pairs are ordered, so eight by one and two by four are distinct answers, and Example 2 lists all four.
"A single number in brackets is not really a matrix." It is, with order one by one, and the chapter opens the whole section with one. It also uses one by one instances when it names the diagonal, scalar and identity matrices two pages later.
"Entries must be numbers." The definition admits functions, and the chapter's third sample matrix is built entirely from expressions in x. What the chapter does restrict, from Part I p. 37 onwards, is that the values be real.
"a sub two three and a sub three two are the same entry." They are different positions in general, and equal only by accident. Section 7 should point at both on the same printed array.
"Constructing a matrix from a rule needs the rule to be linear, or simple." It needs the rule to return one value for each index pair, and nothing more. Example 3's absolute value is not linear and works fine.

Questions to check understanding

Read the order and the entry count off a printed matrix
Name the entry at a stated position, and state the position of a stated entry
List every order a given number of entries permits, and say how many there are
Explain why a prime number of entries permits only two orders
Build a small matrix from a stated rule on the two indices
Write the vertices of a named plane figure as a matrix, in both layouts
Given a two-way table in words, produce the matrix and interpret one named entry back into words
Decide whether two given arrangements of the same data are the same matrix

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

The notebook build-up (§3.2, Part I pp. 34–35). Three steps in order: one count in brackets; then two counts in brackets, with a sentence fixing which is which; then three people and two possessions. The middle step is the one.
The two arrangements (Part I p. 35). Read off the page image, the chapter draws both and annotates both. The first has three rows, one per person, and two columns, notebooks then pens, holding fifteen and six, ten and two, thirteen and five; arrows label the three rows and the two columns. The second has two rows, notebooks then pens, and three columns, one per person; arrows label its two rows and three columns. The same six counts, two orders, two different matrices. These annotated arrays are the closest thing the chapter has to a figure.
The three sample matrices (§3.2, Part I p. 36). A has three rows and two columns and holds minus two and five, zero and root five, three and six. B has three rows and three columns and holds two plus i, three and minus a half; three point five, minus one and two; root three, five and five sevenths. C has two rows and three columns and holds one plus x, x cubed and three; cosine x, sine x plus two, and tangent x. All three are set in square brackets. Together they make the point that an entry can be a whole number, a surd, a decimal, a fraction, a variable expression or a trigonometric function.
The order, and the entry count (§3.2.1, Part I p. 36). The chapter reads the three sample orders as three by two, three by three and two by three, and the entry counts as six, nine and six. Verified: the counts are the products, and the general statement that an m by n matrix holds mn entries follows because each of the m rows contributes n of them.
The general array with subscripts (Part I p. 36). Printed as a full rectangular display with a sub one one at the top left and a sub m n at the bottom right, the order set outside the closing bracket. Beside it the compact form, with the ranges on both indices given as inequalities and both indices declared natural numbers. The chapter then reads one row and one column out of it and states that a sub i j sits in row i and column j, also calling it the (i, j)th element. Read this off the page image: the text layer drops the whole display.
The Note that fixes the chapter's conventions (Part I p. 37). Two items. The first fixes the compact notation with the order attached. The second restricts attention to matrices whose entries are real numbers or real-valued functions. Verified against the page before it: the matrix B printed on Part I p. 36 has two plus i in its top left corner, which is not real. Read closely to be certain of the letter. The example is offered before the restriction is stated, and nothing later in the chapter uses B again, so the two do not collide in any computation — but a student who reads the two pages in order will notice.
Points and vertices as matrices (Part I p. 37). A point in the plane is written either as a column of two or as a row of two, and the chapter does the point with coordinates zero and one both ways. It then takes a quadrilateral with vertices at one and zero, three and two, one and three, and minus one and two, and writes it twice: once as a two by four matrix with the four vertex names printed above the columns, and once as a four by two matrix with the same four names printed down the left of the rows. The chapter draws no picture of the quadrilateral — checked on the printed page — so the explanation supplying one is adding something the page does not have.
Example 1 (Part I pp. 37–38). A headcount split by sex across three factories: thirty and twenty-five, twenty-five and thirty-one, twenty-seven and twenty-six, to be written as a three by two matrix. The second half of the question is the better half — it asks what the entry in row three, column two means, and the answer is a sentence about people, not a number. Verified: that entry is twenty-six, the women workers in the third factory.
Example 2 (Part I p. 38). A matrix with eight entries: which orders are possible? Verified: the ordered pairs of natural numbers multiplying to eight are one and eight, eight and one, four and two, two and four, so four orders. The word ordered is the whole content — four and two is a different shape from two and four.
Example 3 (Part I p. 38). Build a three by two matrix whose entry in row i, column j is half the absolute value of i minus three j. Verified entry by entry: one and five halves in the first row, one half and two in the second, zero and three halves in the third. Note that the rule is evaluated six times and each evaluation is independent of the others — this is the cleanest illustration in the chapter that a matrix is a function of two indices.
Exercise 3.1 Q1 (Part I p. 42). A printed three by four matrix holding two, five, nineteen and minus seven; thirty-five, minus two, five halves and twelve; root three, one, minus five and seventeen. Verified: the order is three by four, the entry count is twelve, and the five requested entries are nineteen, thirty-five, minus five, twelve and five halves. Note that the item asks for a sub two four before a sub two three, so a student who reads the list in printed order has to hold the convention rather than pattern-match.
Exercise 3.1 Q2 and Q3 (Part I p. 42). Twenty-four entries, then thirteen; eighteen entries, then five. Verified: twenty-four admits eight orders, from one by twenty-four down to twenty-four by one through two, three, four, six, eight and twelve rows; eighteen admits six; and thirteen and five, being prime, admit two each. The pairing of a composite with a prime in each item is the point of both.
Exercise 3.1 Q4 and Q5 (Part I p. 42). Q4 builds three two by two matrices from three rules; Q5 builds two three by four matrices from two more. Verified, Q4: the first rule gives two and nine halves in the top row, nine halves and eight below; the second gives one and one half above, two and one below; the third gives nine halves and twenty-five halves above, eight and eighteen below. Q5: the second rule, twice i minus j, gives one, zero, minus one and minus two; then three, two, one and zero; then five, four, three and two. The first rule of Q5 is worth showing beside Example 3 — it is the same absolute-value shape with the sign of the i term moved, and the entries come out different.

Figures to have open

Redraws of the two annotated arrays on Part I p. 35, with the row arrows and column arrows the chapter prints. These are the chapter's own and they carry sections 2 and 3.
A general array with double subscripts for sections 7 and 8, drawn once and reused, with a row strip and a column strip that can be lit independently. Build it with the repo's DataTable component.
A drawing of the quadrilateral with vertices at one and zero, three and two, one and three, and minus one and two, for section 11. This is an addition made here — the chapter names the figure and does not draw it, confirmed on the page image of Part I p. 37.
A caption card for the chapter opening (Part I p. 34): a QR code marked with the Part I catalogue number and the chapter number, and an epigraph attributed to Cantor about freedom in mathematics. There is no portrait on this opening page — checked on the page image. Do not reproduce the epigraph text; name its author and move on.

Where this sits in the book

NCERT Class 12 Mathematics, Chapter 3 "Matrices", §3.1 Introduction, Part I p. 34
§3.2, the notebook build-up and the two annotated arrangements, Part I pp. 34–35
Definition 1 and the three sample matrices, Part I p. 36
§3.2.1, the order, the general array and the double subscript, Part I p. 36
The Note fixing notation and restricting entries to real values, Part I p. 37
Points and the quadrilateral as matrices, Part I p. 37
Examples 1, 2 and 3, Part I pp. 37–38
Exercise 3.1, questions 1 to 5, Part I p. 42
Summary, the first two bullets, Part I p. 73