Intuition The one core ideaA whole system of linear equations is secretly ONE sentence: "a rule (call it A X B backwards , and this page builds — from nothing — every object (A X B
Everything on the parent page — A X = B X = A − 1 B det A adj ( A ) every one of them from nothing , in the order they depend on each other. If a word below is new, you are in the right place.
Before any equation, we need the object everything is written on.
A matrix is just a rectangular grid of numbers arranged in rows (across) and columns (down). We name it with a capital letter like A ( rows ) × ( columns ) A = [ 2 1 3 − 1 ] is a 2 × 2 matrix (2 rows, 2 columns).
The single number sitting in row i j a ij row first, column second . So a 21
A square matrix is one whose number of rows equals its number of columns (size n × n 2 × 2 3 × 3 2 × 3 not square.
Why we care right now: the identity matrix I A − 1 det A adj ( A ) only for square matrices . A non-square grid has no determinant and no inverse at all. This is why the parent topic always uses "n n A
Intuition Why a grid at all?A grid lets us store the coefficients of many equations in one tidy place, so the whole system becomes a single object we can manipulate with one set of rules. Look at the figure: each equation becomes one row , each variable owns one column . When the count of equations matches the count of unknowns, that grid is square — exactly the case where we can hope to reverse it.
WHY the topic needs it: the coefficient matrix A X B A X = B
A matrix with only one column is special enough to get its own name.
A column vector is a matrix with one column, e.g. X = [ x y ] 2 × 1 arrow starting at the origin and pointing to the coordinates ( x , y )
Intuition Two ways to see the same thingA pair of numbers [ x y ]
WHY the topic needs it: X B X
This is the engine of the whole topic, so we go slow.
Definition Dimension-matching ruleYou may multiply A m × n X n × 1 only when the inner numbers match : the columns of A = n X = n outer size m × 1 A ( m × n ) X ( n × 1 ) ⟶ ( m × 1 ) . undefined — you cannot write it. (For a square A n × n X n × 1 n × 1
Definition Row-times-column (matrix × vector)To multiply matrix A X each row of A , march along it multiplying by the matching entry of X [ a c b d ] [ x y ] = [ a x + b y c x + d y ] .
WHAT we just did: row 1 ( a , b ) ( x , y ) a x + b y c x + d y WHY this pattern: because a x + b y exactly the left side of a linear equation. The multiplication rule was designed so that a matrix times a vector reproduces the left-hand sides of equations.
WHAT IT LOOKS LIKE: feeding an input arrow X A
Definition Matrix × matrix (needed for the identities later)The same rule extends to two full grids: to multiply P m × n Q n × p row i j of the product is row i P j Q m × p n [ a c b d ] [ p r q s ] = [ a p + b r c p + d r a q + b s c q + d s ] . p = 1 A adj ( A ) A − 1 A
Worked example Check the machineWith A = [ 2 1 3 − 1 ] X = [ x y ] A X = [ 2 x + 3 y x − y ] . B = [ 5 1 ] exactly 2 x + 3 y = 5 x − y = 1 A X = B is the system.
Common mistake Multiplying column-firstWhy it feels right: we read left-to-right, so grabbing the column of A The fix: always row of the left matrix, column of the right . Row-first is the rule; column-first gives nonsense.
Definition Identity matrix (square only)The identity matrix I square matrix with 1 0 I = [ 1 0 0 1 ] . I X = X X 1 I
WHY the topic needs it: the whole derivation of X = A − 1 B I X I X X I
Definition Inverse matrix (square only)A − 1 A square matrix that undoes A A A − 1 A − 1 A = I and A A − 1 = I . A − 1 A A A − 1 both be defined and both equal the same I A n × n no inverse — full stop.
Intuition The vending-machine pictureA X B A − 1 B X A A − 1 I
WHY the little − 1 In ordinary numbers, 5 − 1 = 5 1 5 × 5 1 = 1 A − 1 A I 1
WHY the topic needs it: X = A − 1 B
We need one number that tells us whether A − 1 square-only quantity.
Definition Determinant of a1 × 1 2 × 2 determinant is a single number computed from a square matrix. We build it up by size:
1 × 1 det [ a ] = a (We need this base case in §6, because deleting a row and column of a 2 × 2 1 × 1 2 × 2 det [ a c b d ] = a d − b c . larger: an n × n 1 × 1 Determinants .) There is no determinant for a non-square grid.
Geometrically the 2 × 2 signed area of the parallelogram that the two column-arrows of A
Intuition Why area = reversibilityIf the two column arrows point in genuinely different directions, they open up a parallelogram with real area — the machine spreads space out and can be undone. If the arrows fall on the SAME line, the parallelogram has zero area : space got squashed onto a line, many inputs land on the same output, and you can never uniquely reverse it. That flat case is exactly det A = 0 det A = 0 singular (non-reversible); one with det A = 0 non-singular .
WHY the topic needs it: A − 1 = det A 1 adj A det A det A = 0 Determinants for the full story.
To actually compute the inverse we need three linked ideas. (Full detail: Cofactors and Minors and Adjoint and Inverse of a Matrix .)
Definition Minor, cofactor, adjugate
A minor M ij delete row i j . For a 2 × 2 1 × 1
A cofactor C ij = ( − 1 ) i + j M ij
The adjugate adj ( A ) transposed (rows and columns swapped): adj ( A ) ij = C j i
Common mistake Skipping the transposeWhy it feels right: the cofactor grid and the adjugate are the same numbers, so they look identical.
The fix: adj ( A ) flipped across its diagonal . Always transpose; verify with the key identity below.
Definition Consistent vs inconsistent
A system is consistent if it has at least one solution.
It is inconsistent if it has no solution.
When det A = 0 X = A − 1 B
det A = 0 derived , not asserted
Take the system A X = B both sides on the left by adj ( A ) adj ( A ) A X = adj ( A ) B . adj ( A ) A = ( det A ) I ( det A ) X ( det A ) X = adj ( A ) B . det A = 0 zero column O adj ( A ) B = O
If adj ( A ) B = O violated — no X no solution (inconsistent) .
If adj ( A ) B = O X infinitely many solutions, or none (you must confirm by substitution).
Qualification: this ( adj A ) B 2 × 2 3 × 3 Consistency of Linear Systems and Gaussian Elimination .
Read each item as a question, hide the answer, and test yourself.
What is a matrix and how do we label an entry a ij A rectangular grid of numbers; a ij i j
What is a square matrix, and why does it matter here? Rows = columns (n × n
What is a column vector, and what picture goes with it? A one-column matrix; an arrow from the origin to the coordinates it lists.
When is the product A X When the columns of A X
How do you multiply a matrix by a column vector? Take each row of the matrix, multiply term-by-term with the vector, and add — row-of-left times column-of-right.
How do you multiply two matrices? Entry ( i , j ) i j
Why does A X = B Row i X i b i i
What is the identity matrix I A square matrix, 1 0 I X = X
What does A − 1 The reverse machine with A − 1 A = A A − 1 = I
What is det [ a ] 1 × 1 Just a
What does det A 2 × 2 det A = 0 The signed area of the column parallelogram; zero means space is squashed flat (singular, non-reversible).
What is the difference between a cofactor grid and adj ( A ) The adjugate is the cofactor grid transposed (rows and columns swapped).
Why is the 2 × 2 The four cofactors give [ d − b − c a ] [ d − c − b a ]
Why does A adj ( A ) = ( det A ) I Diagonal entries are the cofactor expansion of det A 0
How does ( det A ) X = adj ( A ) B det A = 0 If det A = 0 O adj ( A ) B = O = O
What does "consistent" mean? The system has at least one solution.
