Matrices: Introduction to Matrix Algebra

Matrices - Introduction

• Matrix algebra helps reduce complicated equation systems to simple expressions.
• It is adaptable to systematic mathematical treatment for computers.
• A matrix is a set of numbers in a square or rectangular array enclosed by two brackets.
• Matrices have a specified number of rows and columns.
• The dimensions or size of a matrix are described by two numbers: (rows x columns).
• A matrix is denoted by a bold capital letter, with elements denoted by lowercase letters e.g. matrix [A] with elements aij.

Types of Matrices

• Column Matrix/Vector: The number of column is always 1.
• Row Matrix/Vector: The number of row is always 1.
• Rectangular Matrix: Number of rows is not equal to the number of columns (m ≠ n).
• Square Matrix: Number of rows equals the number of columns.
• The principal or main diagonal of a square matrix is composed of all elements aij for which i = j.
• Diagonal Matrix: A square matrix where all elements are zero except those on the main diagonal (aij = 0 for all i ≠ j).
• Unit or Identity Matrix: A diagonal matrix with ones on the main diagonal (aij = 0 for all i ≠ j and aij = 1 for some or all i = j).
• Null (Zero) Matrix: All elements in the matrix are zero (aij = 0 for all i, j).
• Triangular Matrix: A square matrix where elements above or below the main diagonal are all zero.
• Upper Triangular Matrix: A square matrix whose elements below the main diagonal are all zero (aij = 0 for all i > j)
• Lower Triangular Matrix: A square matrix whose elements above the main diagonal are all zero (aij = 0 for all i < j)
• Scalar Matrix: diagonal matrix whose main diagonal elements are equal to the same scalar.

Matrices - Operations

• Equality of Matrices: Two matrices are equal only when all corresponding elements are equal, and their sizes/dimensions are the same.
• If A = B, then B = A for all matrices A and B.
• If A = B and B = C, then A = C for all matrices A, B, and C.
• Addition and Subtraction of Matrices: The sum or difference of two matrices, A and B, of the same size yields a matrix C of the same size (cij = aij + bij).
• Matrices of different sizes cannot be added or subtracted.
• Commutative Law: A + B = B + A
• Associative Law: A + (B + C) = (A + B) + C = A + B + C
• A + 0 = 0 + A = A
• A + (-A) = 0, where (-A) is the matrix composed of -aij as elements.
• Scalar Multiplication of Matrices: Matrices can be multiplied by a scalar.
• If k is a scalar, then kA = Ak.
• k(A + B) = kA + kB
• (k + g)A = kA + gA
• k(AB) = (kA)B = A(k)B
• k(gA) = (kg)A

Multiplication of Matrices

• The product of two matrices is another matrix.
• Two matrices A and B must be conformable for multiplication to be possible.
• The number of columns of A must equal the number of rows of B.

Rules of Multiplying Matrices

• A x B = C only when columns in A = rows in B.
• B x A is not always the same as A x B.
• a11 x b11 + a12 x b21 + a13 x b31 = c11 - Successive multiplication of row i of A with column j of B
• Assuming matrices A, B, and C are conformable for the operations indicated, the following are true:
• AI = IA = A
• A(BC) = (AB)C = ABC (associative law)
• A(B+C) = AB + AC (first distributive law)
• (A+B)C = AC + BC (second distributive law)

Cautions

• AB is not generally equal to BA; BA may not be conformable
• If AB = 0, neither A nor B necessarily = 0
• If AB = AC, B is not necessarily = C

Transpose of a Matrix

• Interchange rows and columns.
• The dimensions of AT are the reverse of dimensions of A.
• (A+B)T = AT + BT
• (AB)T = BTAT
• (kA)T = kAT
• (AT)T = A

Symmetric Matrices

• A square matrix is symmetric if it is equal to its transpose (A = AT).
• When the original matrix is square, transposition does not affect the elements of the main diagonal.
• The identity matrix (I), a diagonal matrix (D), and a scalar matrix (K) are equal to their transpose since the diagonal is unaffected.

Inverse of a Matrix

• Consider a scalar k, the inverse is 1/k = k^-1
• Division of matrices is not defined since there may be AB = AC while B ≠ C, hence matrix inversion is used.
• The inverse of a square matrix, A, if it exists, is the unique matrix (A^-1) where AA^-1 = A^-1A = I
• (AB)^-1 = B^-1A^-1
• (A^-1)^-1 = A
• (A^T) = (A^-1)^T
• (kA) = 1/k A^-1

Types of Matrix

• A square matrix that has an inverse is called a nonsingular matrix
• A matrix that does not have an inverse is called a singular matrix
• Square matrices have inverses except when the determinant is zero
• When the determinant of a matrix is zero, the matrix is singular

Determinant of a Matrix

• To compute the inverse of a matrix, the determinant is required.
• Each square matrix A has a unit scalar value called the determinant of A, denoted by det A or |A|.
• If A = [A] is a single-element (1x1) matrix, then the determinant is defined as the value of the element i.e. |A| = det A = a11
• The determinant of A = a11c11 + a12c12 (determinant 2 x 2 matrix)
• If A is (n x n), its determinant may be defined in terms of order (n-1) or less.
• Where "n" is the order, which is also the same number length for the rows/columns

Minors

• If A is an n x n matrix and one row and one column are deleted, the resulting matrix (n-1) x (n-1) is a "Submatrix" of A.
• The determinant of such a submatrix is called a minor of A and is designated by m(ij ), where "i" and "j" correspond to the deleted row and column respectively.
• m(ij) is the minor of the element a(ij) in A.
• Delete first row and column from A
• The determinant of the remaining 2 x 2 submatrix is the minor of a₁₁

Cofactors

• The cofactor C(ij ) of an element a(ij ) is defined as C(ij) = (-1)^i+j x m(ij)
• The determinant of an n x n matrix A can now be defined as: |A|= det A = a11c11 + a12c12+ ... + a1nc1n. The determinant of A is therefore the sum of the products of the elements of the first row of A and their corresponding cofactors.
• If A is a two-by-two matrix A = a11a22 - a12a21
• Has cofactors c₁₁ = m₁₁
• and c₁₂ = -m₁₂
• The determinant of a 3 x 3 matrix is |A|=a11 (a22 a33-a23a32)-a12 (a21a33-a23a31) + a13 (a21a32-a22a31)