Matrices

A grid of numbers, treated as one object

An attendance register, a mark-sheet across five subjects for a class of thirty, a seating plan for an exam hall — each is a rectangular grid of numbers. This chapter treats that grid as ONE object, not a scattered collection of entries. A MATRIX packages the whole grid so it can be added, scaled, and multiplied as a single unit, the way a single number is.

Some of that arithmetic feels exactly like plain number arithmetic. Addition works entry by entry and commutes — order does not matter. Scalar multiplication spreads over addition, same as with plain numbers. One rule breaks: matrix multiplication does not commute in general. $A B$ and $B A$ can be totally different matrices. One can even be defined while the other is not.

This chapter builds that arithmetic from scratch. It covers what a matrix actually is, and how to add, scale, and multiply two matrices correctly. It also covers which special shapes come up again and again — identity, zero, symmetric, skew-symmetric — and what it means for a square matrix to have an inverse.

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Order, entries, and named matrix shapes

A matrix is a rectangular array of numbers, laid out in rows and columns. A matrix with $m$ rows and $n$ columns has ORDER $m \times n$ — rows first, then columns. The entry in row $i$, column $j$ is written $a_{i j}$. Note the space between $i$ and $j$ inside the brackets: they are two separate labels, not one merged symbol.

Two matrices are EQUAL only when both conditions hold at once: they share the same order, AND every matching entry agrees. $a_{i j} = b_{i j}$ for every $i$, $j$. Matching entries this way is what lets an equation between two matrices be solved for unknown entries. Match order first, then match entry by entry.

Certain matrix shapes turn up so often in this chapter that they earn their own names. A ROW matrix has exactly one row, order $1 \times n$; a COLUMN matrix has exactly one column, order $m \times 1$. A SQUARE matrix has equal rows and columns, order $n \times n$.

Within a square matrix, three more names nest inside each other. A DIAGONAL matrix has every off-diagonal entry zero: $a_{i j} = 0$ whenever $i \neq j$. A SCALAR matrix is a diagonal matrix whose diagonal entries are all the same number. The IDENTITY matrix $I$ is the scalar matrix whose diagonal entries are all $1$ — for example, $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ at order $2 \times 2$.

*A ZERO matrix, every entry $0$, is the one named shape that is not restricted to square* — it can have any order at all.

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Transpose, symmetric, and skew-symmetric matrices

The TRANSPOSE of a matrix $A$, written $A^T$, swaps its rows and columns: row $i$ of $A$ becomes column $i$ of $A^T$. If $A$ has order $m \times n$, then $A^T$ has order $n \times m$ — the shape itself flips.

A SQUARE matrix $A$ is SYMMETRIC when it equals its own transpose, $A^T = A$. In other words, $a_{i j} = a_{j i}$ for every $i$, $j$ — the matrix mirrors across its main diagonal. $A$ is SKEW-SYMMETRIC when $A^T = -A$ instead, which forces $a_{i j} = -a_{j i}$.

*Setting $i = j$ in that skew-symmetric condition gives $a_{i i} = -a_{i i}$.* That is only possible when $a_{i i} = 0$. Every diagonal entry of a skew-symmetric matrix is forced to be zero, with no exceptions.

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Adding matrices and scaling by a number

Two matrices add ENTRYWISE, and only when they share the same order. Entry $(i,j)$ of $A + B$ equals $a_{i j} + b_{i j}$. *There is no way to add a $2 \times 3$ matrix to a $3 \times 2$ matrix.* The orders must match exactly, position for position, before addition even makes sense.

SCALAR multiplication scales every entry by the same number: for a scalar $k$, entry $(i,j)$ of $k A$ equals $k \cdot a_{i j}$. $k A$ keeps the same order as $A$.

Both behave exactly like plain-number arithmetic. Addition is commutative, $A + B = B + A$, and associative — grouping does not matter. Scalar multiplication spreads over matrix addition. That changes once multiplication of two matrices enters the picture — this chapter turns to it next.

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Row times column

Matrix multiplication is NOT entrywise — this is the one place matrix arithmetic stops looking like plain numbers. The product $A B$ is built row-times-column. Entry $(i,j)$ of $A B$ is the dot product of row $i$ of $A$ with column $j$ of $B$: multiply matching entries and add them.

That row-times-column rule only works when the number of columns of $A$ equals the number of rows of $B$. That shared count is exactly what gets summed over, entry by entry. If $A$ is $m \times n$ and $B$ is $n \times p$, the product $A B$ exists and has order $m \times p$.

*If the inner numbers do not match, $A B$ is simply undefined — no matter how $B A$ behaves.* Checking that the sizes match comes first, before a single entry is multiplied.

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What survives, what doesn’t

Two familiar rules survive the move to matrix multiplication. Associativity: $(A B)C = A(B C)$, so a chain of products can be grouped either way without changing the answer. Distributivity: $A(B+C) = A B + A C$, and $(A+B)C = A C + B C$ — multiplication spreads over addition exactly as expected.

The rule that does NOT survive is commutativity: in general $A B \neq B A$. The two products can be completely different, or one can be defined while the other is not. The order of a matrix product is never optional, and it can never be swapped out of habit the way a plain-number multiplication can.

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A B is not B A

Plain numbers multiply in either order without changing the answer. That habit is hard to shake. It invites treating $A B$ and $B A$ as the same matrix too, just written two ways.

Take $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}$. Computing both products entry by entry gives $A B = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$ but $B A = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$ — two different matrices, not the same product read two ways.

Multiplication order is never optional. $A B$ and $B A$ can disagree outright, or one can even be undefined while the other is not. The matching condition from the last section is checked on its own for each order — never assumed to hold both ways at once.

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Invertible matrices

A square matrix $A$ is INVERTIBLE when some matrix $B$ of the same order satisfies $A B = B A = I$. That $B$ is called the INVERSE of $A$, written $A^{-1}$. Not every square matrix has one — a matrix with no inverse is called SINGULAR.

When an inverse exists, it is UNIQUE. Suppose both $B$ and $C$ satisfy $A B = B A = I$ and $A C = C A = I$. Then $B = B I = B(A C) = (B A)C = I C = C$. That chain of equal steps forces $B$ and $C$ to be the same matrix — a matrix can never have two different inverses.

This chapter states only the definition and the uniqueness fact. The METHOD for actually computing $A^{-1}$, via the adjoint, belongs to the next chapter.

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Symmetric plus skew-symmetric

Any square matrix $A$ can be written as a sum of a symmetric matrix and a skew-symmetric matrix, in exactly one way. Define $P = (1/2)(A + A^T)$ and $Q = (1/2)(A - A^T)$; adding them back gives $P + Q = A$ by construction.

$P$ is symmetric: $P^T = (1/2)(A^T + A) = P$, since transpose distributes over addition and $(A^T)^T = A$. $Q$ is skew-symmetric by the same kind of check: $Q^T = (1/2)(A^T - A) = -Q$.

Every square matrix, however messy, splits into one symmetric piece and one skew-symmetric piece that add back to it. This is not a special property of tidy matrices. The formulas for $P$ and $Q$ work on any square matrix at all.

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A B = 0 does not force A = 0 or B = 0

With plain numbers, a product equal to zero forces one of the factors to be zero. That habit invites assuming $A B = 0$ must force $A = 0$ or $B = 0$ too.

Take $A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ -1 & -1 \end{pmatrix}$, neither one the zero matrix. Their product comes out $A B = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ anyway — matrices can act as ZERO DIVISORS, a behaviour plain numbers never show.

*$A B = 0$ only says the product is zero.* It says nothing about the factors on their own. A matrix product can vanish even when both factors are packed with nonzero entries.

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One object, its own arithmetic

A matrix packages a whole grid of numbers into one object with its own arithmetic. Addition and scalar multiplication behave exactly like plain numbers, entrywise. Nothing about this chapter’s first two operations should surprise you.

Multiplication does not. It only works when the first matrix’s column count matches the second matrix’s row count. Even when both $A B$ and $B A$ are defined, they can be different matrices. *Never assume $A B$ equals $B A$, and never assume $A B = 0$ forces $A = 0$ or $B = 0$.* Both habits carry over from plain numbers, and both fail here.

Named shapes — identity, zero, symmetric, skew-symmetric — come up again and again because they simplify this arithmetic. An invertible matrix’s inverse, when it exists, is unique. And every square matrix, however untidy, splits cleanly into a symmetric part and a skew-symmetric part that add back to it exactly.

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Practice set

Exercise 3.1 practises reading off a matrix’s order and entries, and building a matrix from a given rule. It covers naming the types — row, column, square, diagonal, scalar, identity, zero — and uses matrix equality to solve for unknown entries.

Exercise 3.2 practises matrix addition, scalar multiplication, and matrix multiplication on concrete matrices. It checks that the matrix sizes match before multiplying, and uses the associative and distributive properties. *It also tests whether $A B = B A$ each time, since it usually does not.*

Exercise 3.3 practises finding a matrix’s transpose, and testing whether a given square matrix is symmetric or skew-symmetric. It also has students decompose a square matrix into its symmetric and skew-symmetric parts and verify the two pieces add back to the original.

Exercise 3.4 practises checking whether a given pair of square matrices satisfies $A B = B A = I$, which is what invertibility actually means. It also has students use the uniqueness fact to reason about candidate inverses, without yet computing an inverse from scratch — that method belongs to the next chapter.

The Miscellaneous Exercise mixes every idea in the chapter into single multi-step problems, rather than drilling one skill at a time. A typical problem might ask for an order check before multiplying, a transpose, and a symmetric/skew-symmetric split. It might also ask whether a stated product could possibly equal the zero matrix without either factor being zero.