Kuzee
Linear Algebra

Basic Concepts of Matrices

The Definition of a Matrix

Matrix

A matrix is a rectangular array of numbers, usually denoted by a capital letter. An m×nm \times n matrix can be written as:

A=(a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn)=(aij)m×nA = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix} = (a_{ij})_{m \times n}

where aija_{ij} denotes the entry in row ii, column jj.

Types of Matrices

1. Square Matrices

Square Matrix

A matrix whose number of rows equals its number of columns is called a square matrix, that is, m=nm = n.

Example: (1234),(123456789)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}

2. The Zero Matrix

Zero Matrix

A matrix whose entries are all zero is called the zero matrix, written OO or 00.

Example: O=(0000)O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

3. The Identity Matrix

Identity Matrix

A square matrix whose entries on the main diagonal are 1 and whose other entries are 0 is called the identity matrix, written II or EE.

Example: I=(100010001)I = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

4. Diagonal Matrices

Diagonal Matrix

A square matrix whose entries other than those on the main diagonal are all zero is called a diagonal matrix.

Example: (200030004)\begin{pmatrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{pmatrix}

5. Symmetric Matrices

Symmetric Matrix

A square matrix satisfying AT=AA^T = A is called a symmetric matrix.

Example: (123245356)\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix}

6. Skew-Symmetric Matrices

Skew-Symmetric Matrix

A square matrix satisfying AT=−AA^T = -A is called a skew-symmetric matrix.

Example: (02−3−2053−50)\begin{pmatrix} 0 & 2 & -3 \\ -2 & 0 & 5 \\ 3 & -5 & 0 \end{pmatrix}

7. Upper-Triangular and Lower-Triangular Matrices

Upper-Triangular Matrix

A square matrix whose entries below (above) the main diagonal are all zero is called an upper-triangular matrix (a lower-triangular matrix).

Example: An upper-triangular matrix: (123045006)\begin{pmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 0 & 0 & 6 \end{pmatrix}

A lower-triangular matrix: (100240356)\begin{pmatrix} 1 & 0 & 0 \\ 2 & 4 & 0 \\ 3 & 5 & 6 \end{pmatrix}

The Basic Operations on Matrices

1. Matrix Addition

A+B=(aij+bij)m×nA + B = (a_{ij} + b_{ij})_{m \times n}

Example: (1234)+(5678)=(681012)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix}

2. Multiplication by a Scalar

kA=(kaij)m×nkA = (ka_{ij})_{m \times n}

Example: 2(1234)=(2468)2 \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}

3. Matrix Multiplication

(AB)ij=∑k=1naikbkj(AB)_{ij} = \sum_{k=1}^n a_{ik}b_{kj}

Example: (1234)(5678)=(1×5+2×71×6+2×83×5+4×73×6+4×8)=(19224350)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 1\times5+2\times7 & 1\times6+2\times8 \\ 3\times5+4\times7 & 3\times6+4\times8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

4. Transposition of a Matrix

AT=(aji)n×mA^T = (a_{ji})_{n \times m}

Example: (123456)T=(142536)\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}

Properties of Matrix Operations

Properties of Addition

  • Commutativity: A+B=B+AA + B = B + A
  • Associativity: (A+B)+C=A+(B+C)(A + B) + C = A + (B + C)
  • Zero matrix: A+O=O+A=AA + O = O + A = A

Properties of Multiplication by a Scalar

  • Distributivity: k(A+B)=kA+kBk(A + B) = kA + kB
  • Associativity: (kl)A=k(lA)(kl)A = k(lA)
  • Identity element: 1⋅A=A1 \cdot A = A

Properties of Multiplication

  • Associativity: (AB)C=A(BC)(AB)C = A(BC)
  • Distributivity: A(B+C)=AB+ACA(B + C) = AB + AC, (A+B)C=AC+BC(A + B)C = AC + BC
  • Identity matrix: AI=IA=AAI = IA = A
  • Zero matrix: AO=OA=OAO = OA = O

Note: matrix multiplication is not commutative, that is, in general AB≠BAAB \neq BA.

Properties of Transposition

  • (AT)T=A(A^T)^T = A
  • (A+B)T=AT+BT(A + B)^T = A^T + B^T
  • (kA)T=kAT(kA)^T = kA^T
  • (AB)T=BTAT(AB)^T = B^T A^T

Exercises

Exercise 1

Classify the following matrices:

  • A=(1001)A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
  • B=(01−10)B = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}
  • C=(123245356)C = \begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 5 \\ 3 & 5 & 6 \end{pmatrix}
matrixtypes of matricesmatrix operationssymmetric matrix

Approach: Classify each matrix according to the definitions.

Steps:

  1. The matrix AA:

    • It is a square matrix (2×2)
    • Its entries on the main diagonal are 1 and the rest are zero, so it is the identity matrix
    • It satisfies AT=AA^T = A, so it is a symmetric matrix

  2. The matrix BB:

    • It is a square matrix (2×2)
    • All the entries on its main diagonal are zero
    • It satisfies BT=−BB^T = -B, so it is a skew-symmetric matrix

  3. The matrix CC:

    • It is a square matrix (3×3)
    • It satisfies CT=CC^T = C, so it is a symmetric matrix

Answer: AA is the identity matrix and a symmetric matrix, BB is a skew-symmetric matrix, and CC is a symmetric matrix.

Exercise 2

Compute A+BA + B, where: A=(1234),B=(4321)A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 3 \\ 2 & 1 \end{pmatrix}

matrixtypes of matricesmatrix operationsmatrix addition

Approach: Use the definition of matrix addition, adding corresponding entries.

Steps:

A+B=(1+42+33+24+1)=(5555)A + B = \begin{pmatrix} 1+4 & 2+3 \\ 3+2 & 4+1 \end{pmatrix} = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}

Answer: A+B=(5555)A + B = \begin{pmatrix} 5 & 5 \\ 5 & 5 \end{pmatrix}

Exercise 3

Compute 2A−3B2A - 3B, where: A=(1234),B=(0110)A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

matrixtypes of matricesmatrix operationsmultiplication by a scalar

Approach: First compute the scalar multiples, then carry out the subtraction.

Steps:

2A=(2468),3B=(0330)2A = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix}, \quad 3B = \begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix}

2A−3B=(2468)−(0330)=(2138)2A - 3B = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} - \begin{pmatrix} 0 & 3 \\ 3 & 0 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 3 & 8 \end{pmatrix}

Answer: 2A−3B=(2138)2A - 3B = \begin{pmatrix} 2 & 1 \\ 3 & 8 \end{pmatrix}

Exercise 4

Compute ABAB and BABA, where: A=(1201),B=(2013)A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}

matrixtypes of matricesmatrix operationsmatrix multiplication

Approach: Use the definition of matrix multiplication, and verify that matrix multiplication is not commutative.

Steps:

AB=(1201)(2013)=(1×2+2×11×0+2×30×2+1×10×0+1×3)=(4613)AB = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 1\times2+2\times1 & 1\times0+2\times3 \\ 0\times2+1\times1 & 0\times0+1\times3 \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 1 & 3 \end{pmatrix}

BA=(2013)(1201)=(2×1+0×02×2+0×11×1+3×01×2+3×1)=(2415)BA = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 2\times1+0\times0 & 2\times2+0\times1 \\ 1\times1+3\times0 & 1\times2+3\times1 \end{pmatrix} = \begin{pmatrix} 2 & 4 \\ 1 & 5 \end{pmatrix}

Answer: AB=(4613)AB = \begin{pmatrix} 4 & 6 \\ 1 & 3 \end{pmatrix}, BA=(2415)BA = \begin{pmatrix} 2 & 4 \\ 1 & 5 \end{pmatrix}

Note that AB≠BAAB \neq BA, which verifies that matrix multiplication is not commutative.

Exercise 5

Verify that (AB)T=BTAT(AB)^T = B^T A^T, where: A=(1234),B=(5678)A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}

matrixtypes of matricesmatrix operationstranspose

Approach: Compute (AB)T(AB)^T and BTATB^T A^T separately and verify that they are equal.

Steps:

  1. Compute ABAB: AB=(1234)(5678)=(19224350)AB = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix}

  2. Compute (AB)T(AB)^T: (AB)T=(19432250)(AB)^T = \begin{pmatrix} 19 & 43 \\ 22 & 50 \end{pmatrix}

  3. Compute ATA^T and BTB^T: AT=(1324),BT=(5768)A^T = \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}, \quad B^T = \begin{pmatrix} 5 & 7 \\ 6 & 8 \end{pmatrix}

  4. Compute BTATB^T A^T: BTAT=(5768)(1324)=(19432250)B^T A^T = \begin{pmatrix} 5 & 7 \\ 6 & 8 \end{pmatrix} \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} = \begin{pmatrix} 19 & 43 \\ 22 & 50 \end{pmatrix}

Answer: (AB)T=BTAT=(19432250)(AB)^T = B^T A^T = \begin{pmatrix} 19 & 43 \\ 22 & 50 \end{pmatrix}, which verifies the property of transposition.

Tip: the basic concepts of matrices are the foundation of linear algebra, so you should master the various types of matrices and the basic operations.

Summary

Symbols Used in This Article

SymbolTypePronunciation / NoteMeaning in This Article
AAMathematical symbolMatrix AStands for a matrix
IIMathematical symbolIdentity matrixStands for the identity matrix
OOMathematical symbolZero matrixStands for the zero matrix
ATA^TMathematical symbolTranspose of AStands for the transpose of the matrix AA
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