Matrices
The rectangular powerhouse of linear algebra โ organizing data, solving systems, and transforming space
In this chapter
3.1 What is a Matrix?
"The essence of Mathematics lies in its freedom." โ Georg Cantor
A matrix is an ordered rectangular array of numbers or functions, arranged in rows and columns and enclosed within square brackets [ ]. It is one of the most powerful tools in mathematics โ used to solve systems of linear equations, represent transformations, manage data in spreadsheets, and much more.
A matrix having m rows and n columns is called a matrix of order m ร n. It is denoted as A = [aแตขโฑผ]โโโ, where aแตขโฑผ is the element in the ith row and jth column.
A = [aแตขโฑผ]โโโ means A is an m ร n matrix
aแตขโฑผ is the (i, j)th element โ ith row, jth column
m ร n is the "order" or "size" of the matrix
Total elements = m ร n
Matrices can represent data systematically. For example, the number of notebooks and pens possessed by three people can be displayed as a 3 ร 2 matrix where each row represents a person and each column represents an item type.
โข We consider only matrices whose elements are real numbers or functions taking real values
โข A point (x, y) in a plane can be represented as a column matrix [x; y] or row matrix [x, y]
โข The vertices of a closed rectilinear figure can be expressed as a matrix
3.2 Types of Matrices
Matrices are classified based on their structure and element values. Understanding these types is essential for working with matrix operations.
| Type | Definition | Example |
|---|---|---|
| Column Matrix | Only one column (order m ร 1) | [3; โ1; 5] |
| Row Matrix | Only one row (order 1 ร n) | [2, โ3, 7] |
| Square Matrix | Number of rows = number of columns (m = n) | 2ร2, 3ร3, nรn |
| Diagonal Matrix | Square matrix with aแตขโฑผ = 0 when i โ j | [2, 0; 0, โ1] |
| Scalar Matrix | Diagonal matrix with all diagonal elements equal | [3, 0; 0, 3] |
| Identity Matrix (I) | Scalar matrix with all diagonal elements = 1 | [1, 0; 0, 1] |
| Zero Matrix (O) | All elements are zero | [0, 0; 0, 0] |
Every identity matrix is a scalar matrix (with k = 1), but not every scalar matrix is an identity matrix.
Every scalar matrix is a diagonal matrix, but not every diagonal matrix is a scalar matrix.
Every diagonal matrix is a square matrix, but not every square matrix is diagonal.
Equality of Matrices
Two matrices A = [aแตขโฑผ] and B = [bแตขโฑผ] are equal if and only if:
(i) They are of the same order
(ii) Each element of A equals the corresponding element of B, i.e., aแตขโฑผ = bแตขโฑผ for all i, j
3.3 Matrix Operations
Addition of Matrices
Two matrices can be added only if they have the same order. The sum is obtained by adding corresponding elements.
A + B = [aแตขโฑผ + bแตขโฑผ] = C, where cแตขโฑผ = aแตขโฑผ + bแตขโฑผ
Scalar Multiplication
Multiplying a matrix by a scalar k means multiplying every element of the matrix by k.
If A = [aแตขโฑผ]โโโ and k is a scalar, then:
kA = [k ยท aแตขโฑผ]โโโ
Example: If A = [3, 1; โ5, 7], then 2A = [6, 2; โ10, 14]
The negative of a matrix is defined as โA = (โ1)A, and the difference is A โ B = A + (โ1)B.
Properties of Matrix Addition
Commutative Law
A + B = B + A
Associative Law
(A + B) + C = A + (B + C)
Additive Identity
A + O = A (zero matrix)
Additive Inverse
A + (โA) = O
Properties of Scalar Multiplication
For matrices A, B of same order and scalars k, l:
(i) k(A + B) = kA + kB
(ii) (k + l)A = kA + lA
3.4 Matrix Multiplication
Matrix multiplication is fundamentally different from addition. It is not element-wise โ instead, it uses a row-by-column dot product.
The product AB is defined only when the number of columns in A equals the number of rows in B.
If A is m ร n and B is n ร p, then AB is m ร p.
Matrix multiplication is NOT commutative: AB โ BA in general!
where cแตขโ = aแตขโbโโ + aแตขโbโโ + ... + aแตขโbโโ = ฮฃโฑผโโโฟ aแตขโฑผbโฑผโ
Why AB โ BA
Even when both AB and BA are defined, they are generally not equal. Moreover, AB and BA may have different orders!
Example: If A is 2ร3 and B is 3ร2, then AB is 2ร2 while BA is 3ร3.
Exception: Diagonal matrices of the same order commute: AB = BA.
Properties of Matrix Multiplication
Associative Law
(AB)C = A(BC)
Distributive Laws
A(B+C) = AB + AC
(A+B)C = AC + BC
Multiplicative Identity
IA = AI = A
Zero Product
AB = O does NOT imply A = O or B = O
For real numbers, if ab = 0 then either a = 0 or b = 0. This is NOT true for matrices! Two non-zero matrices can multiply to give a zero matrix.
3.5 Transpose of a Matrix
The transpose of a matrix is obtained by interchanging its rows and columns. If A is an m ร n matrix, its transpose A' (or Aแต) is an n ร m matrix.
The (i, j)th element of A' = the (j, i)th element of A
Properties of Transpose
| Property | Statement | Meaning |
|---|---|---|
| Double Transpose | (A')' = A | Transposing twice returns original |
| Scalar Multiple | (kA)' = kA' | Scalar factors come out |
| Addition | (A + B)' = A' + B' | Transpose distributes over addition |
| Multiplication | (AB)' = B'A' | Order reverses! |
(AB)' = B'A', not A'B'! The order of multiplication reverses when taking the transpose of a product.
3.6 Symmetric & Skew-Symmetric Matrices
These special types of square matrices have a beautiful relationship with the transpose operation.
Symmetric Matrix
A' = A, meaning aแตขโฑผ = aโฑผแตข for all i, j. The matrix mirrors about its main diagonal.
Skew-Symmetric Matrix
A' = โA, meaning aแตขโฑผ = โaโฑผแตข. All diagonal elements must be 0.
Theorems
For any square matrix A with real entries:
A + A' is always a symmetric matrix
A โ A' is always a skew-symmetric matrix
Any square matrix can be expressed as the sum of a symmetric and a skew-symmetric matrix:
A = ยฝ(A + A') + ยฝ(A โ A')
where ยฝ(A + A') is symmetric and ยฝ(A โ A') is skew-symmetric.
3.7 Elementary Operations
There are six elementary operations (transformations) on a matrix โ three for rows and three for columns. These are fundamental for finding inverses and solving systems.
| Operation | Row Version | Column Version | Description |
|---|---|---|---|
| Type 1 | Rแตข โ Rโฑผ | Cแตข โ Cโฑผ | Interchange two rows (or columns) |
| Type 2 | Rแตข โ kRแตข (k โ 0) | Cแตข โ kCแตข (k โ 0) | Multiply a row (or column) by a non-zero scalar |
| Type 3 | Rแตข โ Rแตข + kRโฑผ | Cแตข โ Cแตข + kCโฑผ | Add a multiple of one row (or column) to another |
Starting with A = [1, 2; 2, 1]:
Row 2 becomes: [2, 1] โ 2ร[1, 2] = [2โ2, 1โ4] = [0, โ3]
Result: [1, 2; 0, โ3] โ This is the first step in converting to row echelon form!
3.8 Invertible Matrices
A square matrix A is invertible if there exists another square matrix B of the same order such that AB = BA = I (the identity matrix). We write B = Aโปยน.
A square matrix A of order m is invertible if there exists B such that:
AB = BA = Iโ
Then B is the inverse of A, denoted Aโปยน, and A is the inverse of B.
Key Theorems
Uniqueness
Inverse of a square matrix, if it exists, is unique.
Reversal Law
(AB)โปยน = BโปยนAโปยน โ order reverses!
Finding Inverse Using Elementary Operations
To find Aโปยน using row operations:
Step 1: Write A = IA
Step 2: Apply row operations on both sides until LHS becomes I
Step 3: When LHS = I, RHS = Aโปยน
If any row becomes all zeros during the process, Aโปยน does not exist.
โข A rectangular matrix does not have an inverse
โข If B is the inverse of A, then A is also the inverse of B
โข Inverse is unique โ there cannot be two different inverses for the same matrix
3.9 Summary
โข A matrix is an ordered rectangular array of numbers of order m ร n
โข Seven types: column, row, square, diagonal, scalar, identity, zero
โข Addition: same order, element-wise โ commutative and associative
โข Scalar multiplication: multiply every element by the scalar
โข Multiplication: row ร column dot product โ NOT commutative (AB โ BA)
โข Transpose: interchange rows and columns โ (AB)' = B'A'
โข Symmetric: A' = A ยท Skew-symmetric: A' = โA (diagonal = 0)
โข Any square matrix = symmetric + skew-symmetric
โข 6 elementary operations: interchange, scale, add multiple
โข Inverse: AB = BA = I โ B = Aโปยน (unique if exists)