A matrix is a rectangle of numbers in which position carries meaning. Once numbers sit in agreed places, we can add whole tables at once, scale them, multiply them to make bills, flip them in a mirror and even undo them. Computer graphics, spreadsheets, search engines and codes all run on this idea.
A matrix is a table of numbers in brackets. Where a number sits tells you what it means.
A matrix is an ordered rectangular array of numbers (or functions), written in brackets. Each number is an entry (or element). This one is Example 1’s table of workers.
Horizontal lines are rows, vertical lines are columns. Always rows first. A matrix of order m × n holds m · n entries.
aij is the entry in row i and column j. In the matrix above, a32 = 26.
Row 2 is the second line from the top. In a table of factories, each row is one factory.
Column 3 is the third line from the left, like columns holding up a building.
A 2 × 3 matrix has 2 rows and 3 columns. A 3 × 2 matrix is a different shape.
a21 is in row 2, column 1. a12 is somewhere else.
A point (x, y) can be written as a column [x; y] or a row [x y]. List the corners of a shape as columns and the whole shape becomes one matrix. The book’s quadrilateral A(1, 0), B(3, 2), C(1, 3), D(−1, 2) becomes a 2 × 4 matrix, or a 4 × 2 matrix if we use rows.
Men and women workers in factories I, II, III: 30 and 25, 25 and 31, 27 and 26. Write a 3 × 2 matrix. What is the entry in row 3, column 2?
A matrix has 8 elements. What orders are possible?
Construct a 3 × 2 matrix with aij = ½|i − 3j|.
Some matrices come up so often that they have names. Some names describe the shape; others describe a pattern inside a square matrix.
Column, row and square matrices are named after their shape.
In a square matrix, the entries running from the top-left corner to the bottom-right. The pattern names (diagonal, scalar, identity) all depend on it.
Only one column, like a shopping list or a point [x; y].
Only one row, like a price list [5 50].
As many rows as columns. Only square matrices have a main diagonal.
Square, and everything off the main diagonal is 0.
A diagonal matrix whose diagonal entries are all the same number k. It acts like the ordinary number k.
1s on the diagonal, 0s elsewhere. It works like the number 1: AI = IA = A.
Any shape. It works like the number 0: A + O = A.
Every identity matrix is a scalar matrix, and every scalar matrix is a diagonal matrix, and every diagonal matrix is square. The names get more demanding, like nested boxes. A 1 × 1 matrix such as [4] is all of column, row, square, diagonal and scalar at once.
Two matrices are equal only when they have the same shape and the same number in every place.
A = B only when (i) they have the same order and (ii) every pair of corresponding entries is equal.
So one matrix equation between two 3 × 3 matrices secretly contains nine ordinary equations, one for each position.
Same order (2 × 2) and every position agrees.
Same numbers, but 3 and 2 have swapped places. Position matters.
Find a, b, c, x, y, z when [x + 3 z + 4 2y − 7; −6 a − 1 0; b − 3 −21 0] = [0 6 3y − 2; −6 −3 2c + 2; 2b + 4 −21 0].
Find a, b, c, d from [2a + b a − 2b; 5c − d 4c + 3d] = [4 −3; 11 24].
To add two matrices, add the numbers that sit in the same place. To scale one, multiply every number by the same amount.
Fatima has two factories, A and B, each making boys’ and girls’ shoes in three price categories. The total in each category adds the numbers in the same place. Defined only for matrices of the same order.
If factory A doubles its output, every entry doubles. That is a scalar multiple.
Taking away B is adding its negative.
Each entry is aij + bij = bij + aij, because numbers commute.
Brackets can move, entry by entry.
The zero matrix of the same order changes nothing.
−A undoes A.
and (k + l)A = kA + lA. Scaling spreads over addition.
A = [√3 1 −1; 2 3 0], B = [2 √5 1; −2 3 ½]. Find A + B.
A = [1 2 3; 2 3 1], B = [3 −1 3; −1 0 2]. Find 2A − B.
A = [8 0; 4 −2; 3 6], B = [2 −2; 4 2; −5 1]. Find X with 2A + 3X = 5B.
X + Y = [5 2; 0 9], X − Y = [3 6; 0 −1]. Find X and Y.
2[x 5; 7 y − 3] + [3 −4; 1 2] = [7 6; 15 14]. Find x, y.
Two farmers’ September (A) and October (B) rice sales. Find the combined sales, the decrease, and the 2% profit on October sales.
To multiply matrices, every row meets every column. Multiply matching items and add, just like working out a shopping bill.
Meera buys 2 pens and 5 story books. Pens cost ₹5 and books ₹50, so she pays ₹260. Her bill is a row of quantities times the column of prices.
Nadeem buys 8 pens and 10 books: 8 · 5 + 10 · 50 = ₹540. A second shop (₹4 and ₹40) adds a second price column, and all four bills form one matrix.
If A is m × n and B is n × p, then AB is m × p. The columns of A must match the rows of B.
The (i, k) entry of AB: row i of A meets column k of B.
1. AB can exist while BA does not. A 2 × 3 times a 3 × 3 is fine, but 3 × 3 times 2 × 3 is not.
2. Even when both exist, AB ≠ BA in general. Order matters, like "put on socks, then shoes". (Diagonal matrices of the same order are an exception: they always commute.)
3. AB = O does not force A = O or B = O. [0 −1; 0 2][3 5; 0 0] = O, though neither factor is zero.
A = [6 9; 2 3], B = [2 6 0; 7 9 8]. Find AB.
A = [1 −2 3; −4 2 5], B = [2 3; 4 5; 2 1]. Find AB and BA.
A = [1 0; 0 −1], B = [0 1; 1 0]. Show AB ≠ BA.
A = [0 −1; 0 2], B = [3 5; 0 0]. Find AB.
Matrix multiplication keeps most of the rules of ordinary multiplication, except that you may never swap the order of the factors.
Brackets can move. So ABC has one meaning.
and (A + B)C = AC + BC, keeping each matrix on its own side.
For every square matrix A, the identity of the same order changes nothing.
Always keep the order you were given.
Show (AB)C = A(BC) for A (3 × 3), B (3 × 2), C (2 × 4).
Verify (A + B)C = AC + BC for the given 3 × 3 matrices and C = [2; −2; 3].
A = [1 2 3; 3 −2 1; 4 2 1]. Show A³ − 23A − 40I = O.
Cost per contact 40, 100, 50 paise; contacts in cities X and Y: [1000 500 5000; 3000 1000 10,000]. Total spent?
The transpose swaps rows and columns. Picture every entry flying across the main diagonal to its mirror position.
The transpose A′ (or AT) of an m × n matrix is the n × m matrix you get by swapping rows and columns: row 1 of A becomes column 1 of A′.
The entry in row i, column j of A′ is the entry in row j, column i of A.
Flip twice and you are back.
Scaling and flipping can be done in either order.
Flip the sum, or add the flips.
The order reverses, like taking off socks and shoes.
A = [3 √3 2; 4 2 0], B = [2 −1 2; 1 2 4]. Verify (A′)′ = A, (A + B)′ = A′ + B′, (kB)′ = kB′.
A = [−2; 4; 5], B = [1 3 −6]. Verify (AB)′ = B′A′.
A symmetric matrix looks the same in the diagonal mirror. A skew symmetric matrix turns into its own negative.
A square matrix is symmetric if it equals its transpose. Think of a distance chart between towns.
A square matrix is skew symmetric if its transpose is its negative. Think of a goal-difference table.
Put i = j in the skew rule: every diagonal entry of a skew symmetric matrix is 0. Goal difference against yourself is 0.
(A + A′)′ = A′ + A = A + A′, and (A − A′)′ = A′ − A = −(A − A′).
Every square matrix is a symmetric part plus a skew symmetric part.
Express B = [2 −2 −4; −1 3 4; 1 −2 −3] as symmetric + skew symmetric.
A, B symmetric of the same order. Show AB is symmetric if and only if AB = BA.
An inverse undoes what a matrix does. It is the matrix version of the inverse functions in Chapter 1.
If A is square of order m and a square matrix B of the same order gives AB = BA = I, then B is the inverse of A, written A−1, and A is invertible.
These two multiply to I in both orders. Only square matrices can have inverses, because AB and BA must both exist and be the same size.
An inverse, if it exists, is unique (Theorem 3): if B and C both work, they are the same matrix.
Theorem 4: to undo “A after B”, first undo A, then undo B.
How to find an inverse in general is the subject of the next chapter (Determinants). For a 2 × 2 matrix [a b; c d], the inverse exists exactly when ad − bc ≠ 0.
If A = [cos θ sin θ; −sin θ cos θ], prove Aⁿ = [cos nθ sin nθ; −sin nθ cos nθ].
A = [2 −1; 3 4], B = [5 2; 7 4], C = [2 5; 3 8]. Find D with CD − AB = O.
Every multiple-choice question from the exercises, in one place. Tap an option to check it and see the reasoning.
The questions come from Exercises 3.1–3.4 and the miscellaneous exercise.
A matrix is an ordered rectangular array. Order m × n = m rows, n columns; aij sits in row i, column j.
Column (m × 1), row (1 × n), square (m = n); diagonal, scalar and identity are square with patterns on the diagonal; O is all zeros.
Equal: same order and aij = bij everywhere. Add and scale entry by entry; A − B = A + (−1)B.
AB needs (columns of A) = (rows of B); cik = Σ aijbjk. Associative and distributive, but not commutative, and AB = O is possible with A, B ≠ O.
Transpose swaps rows and columns: (A′)′ = A, (kA)′ = kA′, (A + B)′ = A′ + B′, (AB)′ = B′A′.
Symmetric A′ = A; skew symmetric A′ = −A (zero diagonal). Every square matrix = ½(A + A′) + ½(A − A′).
Inverse: AB = BA = I. Only square matrices can have one, it is unique, and (AB)−1 = B−1A−1.
Each example is a small animated story. Entries light up when they are used, rows meet columns, and numbers fly across the diagonal. Press Next step (or Play) and watch the matrices build the solution. Under the controls, each step is explained in detail, with links to the exact basics it uses. After reading a basic, press ← Back to return to the same step. Most examples also have a Try it panel where you can type your own numbers.
A matrix is a table whose positions carry meaning. Its order is its shape, and every entry has an address aij.
Equal matrices match position by position, so each position gives one equation.
Add or subtract entries that share a position; multiply every entry by the same number. Matrix equations are then solved just like ordinary ones.
Each entry of AB is a row of A times a column of B. Watch why the order matters, why AB can be zero, and how brackets can move.
Transposing swaps rows and columns. Symmetric matrices look the same in the mirror; skew symmetric ones turn into their negatives.
A rotation matrix raised to a power, a symmetric-product proof, and a missing matrix found from four equations.
Every photo on your phone is a matrix. Every bill, report card and league table is one. Video games move characters with 2 × 2 matrices, and secret codes hide messages with them. Here are six scenes where the chapter shows up, each tied to a different idea.
A black-and-white photo is stored as a matrix: one entry per pixel, 0 for black up to 9 for white (real cameras use 0 to 255). Photo apps edit pictures by doing matrix arithmetic on that grid.
Each friend’s order is a row of quantities. Each canteen’s menu is a column of prices. Their product is a table of every possible bill, so each friend can see where to eat.
A school combines theory, practical and project marks with weights. The marks table times the weight column gives every student’s final score at once. Change the weights and watch the ranking change.
A video game stores a sprite as a list of corner points and moves it by multiplying every corner by a 2 × 2 matrix. Doing one move and then another is a matrix product. Pick two moves and try both orders.
G lists the goals each team scored against each other team. It is not symmetric. But G + G′ gives the total goals in each match (symmetric), and G − G′ gives the goal difference (skew symmetric). That is Theorem 1 of the chapter, living in a football league.
Turn letters into numbers (A = 1, …, Z = 26, space = 0), stack them in pairs as columns, and multiply by the key K = [2 3; 1 2]. The result looks like random numbers. Only someone who knows K−1 = [2 −3; −1 2] can multiply back and read it, because K−1K = I.
Decide first, then tap each card to check.
Every numeric question has an animated walkthrough here; the multiple-choice questions are in the Lesson’s self-check. Pick a question from the chips above each player. Press Play to watch it solved, or step through with Next. Each step has a short "why" under it and links to the basics it needs.
Count rows and columns, read entries by their address, build matrices from a rule, and match equal matrices position by position. (Q8–Q10 are in the self-check quiz.)
Addition pairs entries in the same position. Multiplication runs each row of the first matrix along each column of the second.
Treat matrix equations like ordinary ones: move terms across, divide by numbers, then compare entries.
Products that differ when swapped, matrix polynomials, and money problems solved by one multiplication. (Q18 is a proof of the same kind as Q13; Q21–Q22 are in the quiz.)
Transpose by flying every entry across the diagonal, test symmetry, and split any square matrix into a symmetric part and a skew symmetric part. (Q11–Q12 are in the quiz.)
Unknown entries, a matrix sandwich equal to zero, a quadratic in A, market revenue and a missing matrix. (Q1–Q2 are proofs like Example 24; Q9–Q11 are in the quiz.)
Matrices rest on a handful of older ideas: tables, equations, adding, multiplying, mirrors and undoing. Here each one is explained from its roots. You will see not just what the rule is but why it has to be that way. Every example and exercise links back here exactly where you need it, and a button takes you straight back to the problem.
A plain list like 30, 25, 25, 31, 27, 26 means nothing on its own. Is 31 about men or women? Which factory? The moment we arrange the same numbers in a grid, with factories down the side and men and women across the top, each number gets two labels for free: its row and its column. That is the whole reason matrices exist. They are numbers whose position is part of their meaning.
The words come from everyday life. A row of seats in a cinema runs across, from side to side. A column of a building stands up and down. In a matrix, rows are counted from the top and columns from the left.
A 4 × 6 hall and a 6 × 4 hall have the same number of seats (24) but a different shape. If one person said "row 2, seat 5" and another meant "seat 2, row 5", they would sit in different places. So everyone agrees: the order of a matrix is written rows × columns, and the address aij is written row, then column. a25 and a52 are different entries (and in a 4 × 6 matrix, a52 does not even exist).
Multiplication is repeated addition. There are m rows and each row holds n entries, so the total is n + n + … + n (m times) = m × n. That is also why the possible orders of a matrix with 8 entries are the factor pairs of 8: 1 × 8, 2 × 4, 4 × 2 and 8 × 1. A prime number of entries, like 13, only allows one long row or one long column.
In a square grid (as many rows as columns) the cells a11, a22, a33, … run in a straight line from the top-left corner to the bottom-right corner. That line is the main diagonal. On it, the row number equals the column number (i = j). A rectangular grid has no such corner-to-corner line, which is why "diagonal", "scalar" and "identity" matrices must be square.
A matrix is a spreadsheet with the labels removed. The labels are not lost: they are remembered by where each number sits. Row i, column j is a number's home address.
When two matrices are equal, each pair of matching entries gives an ordinary equation. So we need to be good at solving ordinary equations. The root idea is a balance scale: the = sign says both pans weigh the same. Whatever you do to one pan, do the same to the other, and it stays level.
In 2x + 3 = 7, the unknown x was first doubled and then had 3 added. To find x, undo those steps in reverse order: take 3 away from both sides (2x = 4), then halve both sides (x = 2). Each move keeps the balance level.
One equation like a + 4b = −7 has endless solutions: a = 1, b = −2, or a = −7, b = 0, and so on. Each one is a point on a straight line. A second equation is a second line. The answer that satisfies both is where the two lines cross. Two methods find that point without drawing:
Scale them so one unknown has opposite (or equal) amounts, then add (or subtract) so it cancels.
Double 2a + b = 4 to get 4a + 2b = 8.
Add a − 2b = −3: b cancels, leaving 5a = 5.
From 5c − d = 11, get d = 5c − 11.
Put that into 4c + 3d = 24 and only c is left:
19c = 57, so c = 3.
Two lines can be parallel and never cross. Then there is no solution. That is what happens in Exercise 3.1 Q9, where one entry forces x = −7/3 and another forces x = −2/3: no single x works, so the matrices can never be equal.
5 apples + 3 apples = 8 apples. But 5 apples + 3 bananas is not "8 of something". Adding is combining amounts of the same kind. In a matrix, the "kind" of a number is its position: row 2, column 1 might mean "Gurcharan’s Basmati sales". So the only sensible way to add two tables is to add the numbers that sit in the same place.
If two tables have different shapes, some positions have no partner. Nothing sensible can be added there, so the sum is simply not defined.
A recipe for 4 people needs 2 cups of rice, 3 onions and 1 spoon of salt. For 12 people you multiply every ingredient by 3. Nothing moves; each amount just grows. That is exactly kA: every entry multiplied by k. Multiplying by ½ halves everything, and multiplying by −1 flips every sign, which is how we get −A.
On a number line, −3 is the mirror of 3 about 0, and 7 − 3 means 7 + (−3): walk 3 steps back. So A − B = A + (−1)B: flip every sign of B and add position by position. This is why a matrix can have negative entries even when every original number was positive (a sales decrease).
Adding matrices is adding many separate little sums at once, one for each position. Scaling is doing the same multiplication to every entry. Neither ever mixes two different positions.
Matrix multiplication looks strange at first: why not just multiply matching entries, like addition? Because it was invented to answer the most common question in business: what does it all cost?
3 pens at ₹5 each cost 5 + 5 + 5 = 3 × 5 = ₹15. A basket with several items costs (quantity × price) for each item, added up: 3 pens and 2 books at ₹5 and ₹50 cost 3·5 + 2·50 = ₹115. Write the quantities as a row (3, 2) and the prices as a column (5, 50). The bill pairs the first with the first, the second with the second, multiplies each pair and adds. That is row × column.
Put each person’s quantities in a row of the first matrix and each shop’s prices in a column of the second. The bill of person i at shop k is row i × column k, so it goes in row i, column k of the answer. The answer has one row per person and one column per shop.
Every quantity needs its own price. If a row lists 3 items, the column must list 3 prices, no more and no fewer. So the number of columns of A must equal the number of rows of B. Write the orders as dominoes: (m × n)(n × p). The touching numbers must match, and the answer keeps the outside ones: m × p.
Each row of 3 quantities meets each column of 3 prices.
A row has 3 numbers but a column only 2: one item has no price.
In AB the rows come from A and the columns from B. Swap them and different rows meet different columns, so the answer usually changes, and sometimes BA does not even exist. Think of "socks, then shoes" versus "shoes, then socks": the same two actions in a different order give a different result.
Each entry of AB answers one question: "row i of A, paired with column k of B, multiplied and added up: what do you get?" If you can work out a shopping bill, you can multiply matrices.
A school timetable can be drawn with days down the side and periods across the top, or with periods down the side and days across. The information is identical; only the layout is turned. Swapping rows and columns like this is the transpose. Row 1 becomes column 1, row 2 becomes column 2, and so on.
Transposing sends the entry in row i, column j to row j, column i. Picture a mirror along the main diagonal: every entry jumps to its reflection. Entries on the diagonal (i = j) are their own reflection and stay put.
A distance chart between towns is symmetric: the distance from Delhi to Agra equals the distance from Agra to Delhi, so the entry in row "Delhi", column "Agra" equals the entry in row "Agra", column "Delhi". Every entry equals its mirror image: aij = aji, which is exactly A′ = A. The diagonal holds each town’s distance to itself (0), but a symmetric matrix may have any numbers on its diagonal.
A goal-difference table in a league is skew symmetric. If India beat Nepal 3–1, India’s row shows +2 against Nepal and Nepal’s row shows −2 against India: aji = −aij, i.e. A′ = −A. A team’s goal difference against itself must equal its own negative, and the only number equal to its own negative is 0. That is why every diagonal entry of a skew symmetric matrix is 0.
For adding, the number that does nothing is 0 (a + 0 = a), and the undo of adding a is adding its opposite −a. For multiplying, the number that does nothing is 1 (a × 1 = a), and the undo of multiplying by a is multiplying by its reciprocal 1/a, because a × (1/a) = 1. Matrices copy all of this: O does nothing for addition, I does nothing for multiplication, and A−1 undoes A.
Multiplying by 0 sends every number to 0. Once 3 and 5 have both become 0, nothing can tell you which one you started with, so nothing can undo it. An undo is only possible when no information is lost. Matrices can lose information in the same way (squashing the whole plane onto a line), and then they have no inverse either.
I has 1s on the diagonal and 0s elsewhere. When a row of A meets a column of I, the single 1 picks out exactly one entry of the row, so AI = A. Likewise IA = A. It is the "do nothing" of multiplication.
An inverse B must satisfy both AB = I and BA = I, with the same I. If A were 2 × 3, B would have to be 3 × 2, and then AB is 2 × 2 while BA is 3 × 3. They cannot both be the same identity matrix. So both A and B must be square of the same order.
Put on socks, then shoes. To undo, take off the shoes first, then the socks. In the same way (AB)−1 = B−1A−1 and (AB)′ = B′A′: undo the last step first.
Example 23 and a few exercises use two ideas from earlier years. Here they are from the roots.
On a circle of radius 1, the point at angle θ is (cos θ, sin θ). Turning first by B and then by A is a single turn by A + B. Working out where the point lands gives the two compound-angle formulas:
They are exactly the entries you get when you multiply two rotation matrices, which is why rotation matrices "add their angles".
To prove a statement P(n) for every natural number n, it is enough to show two things: (1) P(1) is true: the first domino falls; (2) whenever P(k) is true, P(k + 1) is true: each falling domino knocks over the next. Then P(1) ⇒ P(2) ⇒ P(3) ⇒ … and they all fall. Both parts are needed, as the lab shows.
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