Overview: Matrices | Mathematics for ACT PDF Download

Introduction

• Matrices are rectangular arrays of numbers, symbols, or characters where all of these elements are arranged in each row and column. An array is a collection of items arranged at different locations. 
• Let's envision a scenario where points are positioned in space, each occupying a distinct location. As these points are organized, they form an array referred to as a matrix. Within this matrix, individual components are termed as elements. Each matrix is characterized by a finite number of rows and columns, with each element attributed to specific rows and columns exclusively. The arrangement of rows and columns determines the matrix's order. For instance, if a matrix comprises 3 rows and 2 columns, its order is denoted as 3⨯2.

Definition of Matrices

A matrix is defined as a rectangular array comprising numbers, symbols, or characters. Matrices are distinguished by their order, which is expressed as the number of rows multiplied by the number of columns. A matrix is typically represented as [P]m⨯n, where P denotes the matrix, m represents the number of rows, and n signifies the number of columns. In mathematical contexts, matrices serve as valuable tools for solving a multitude of problems, including linear equations and beyond.

Understanding the Order of a Matrix

The order of a matrix provides insights into the number of rows and columns it encompasses. This order is represented as the product of the number of rows and the number of columns. For instance, if a matrix comprises 4 rows and 5 columns, its order is articulated as 4⨯5. It's crucial to note that the first number in the order signifies the count of rows within the matrix, while the second number indicates the number of columns.

Subtraction of Matrices

Subtraction of Matrices is the difference between the elements of two matrices of the same order to give an equivalent matrix of the same order whose elements are equal to the difference of elements of two matrices. The subtraction of two matrices can be represented in terms of the addition of two matrices. Let’s say we have to subtract matrix B from matrix A then we can write A – B. We can also rewrite it as A + (-B). 

Scalar Multiplication of Matrices

Scalar Multiplication of matrices refers to the multiplication of each term of a matrix with a scalar term. If a scalar let’s ‘k’ is multiplied by a matrix then the equivalent matrix will contain elements equal to the product of the scalar and the element of the original matrix.

Multiplication of Matrices

In the multiplication of matrices, two matrices are multiplied to yield a single equivalent matrix. The multiplication is performed in the manner that the elements of the row of the first matrix multiply with the elements of the columns of the second matrix and the product of elements are added to yield a single element of the equivalent matrix. If a matrix [A]_i⨯j is multiplied with matrix [B]_j⨯k then the product is given as [AB]_i⨯k.

Properties of Matrix Addition and Multiplication

The properties followed by Multiplication and Addition of Matrices is listed below:

• A + B = B + A (Commutative)
• (A + B) + C = A + (B + C) (Associative)
• AB ≠ BA (Not Commutative)
• (AB) C = A (BC) (Associative)
• A (B+C) = AB + AC (Distributive)

Transpose of Matrix

Transpose of Matrix is basically the rearrangement of row elements in column and column elements in a row to yield an equivalent matrix. A matrix in which the elements of the row of the original matrix are arranged in columns or vice versa is called Transpose Matrix. The transpose matrix is represented as A^T. if A = [a_ij]_mxn , then A_T = [b_ij]_nxm where b_ij = a_ji.

Properties of the Transpose of a Matrix

The properties of the transpose of a matrix are mentioned below:

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

Trace of Matrix

Trace of a Matrix is the sum of the diagonal elements of a square matrix. Trace of a matrix is only found in the case of a square matrix because diagonal elements exist only in square matrices.

Types of Matrices

Based on the number of rows and columns present and the special characteristics shown, matrices are classified into various types.

• Row Matrix: A Matrix in which there is only one row and no column is called Row Matrix.
• Column Matrix: A Matrix in which there is only one column and now row is called a Column Matrix.
• Horizontal Matrix: A Matrix in which the number of rows is less than the number of columns is called a Horizontal Matrix.
• Vertical Matrix: A Matrix in which the number of columns is less than the number of rows is called a Vertical Matrix.
• Rectangular Matrix: A Matrix in which the number of rows and columns are unequal is called a Rectangular Matrix.
• Square Matrix: A matrix in which the number of rows and columns are the same is called a Square Matrix.
• Diagonal Matrix: A square matrix in which the non-diagonal elements are zero is called a Diagonal Matrix.
• Zero or Null Matrix: A matrix whose all elements are zero is called a Zero Matrix. A zero matrix is also called as Null Matrix.
• Unit or Identity Matrix: A diagonal matrix whose all diagonal elements are 1 is called a Unit Matrix. A unit matrix is also called an Identity matrix. An identity matrix is represented by I. 
• Symmetric matrix: A square matrix is said to be symmetric if the transpose of the original matrix is equal to its original matrix. i.e. (AT) = A. 
• Skew-symmetric Matrix: A skew-symmetric (or antisymmetric or antimetric[1]) matrix is a square matrix whose transpose equals its negative i.e. (AT) = -A. 
• Orthogonal Matrix: A matrix is said to be orthogonal if AA^T = A^TA = I 
• Idempotent Matrix: A matrix is said to be idempotent if A² = A 
• Involutory Matrix: A matrix is said to be Involutory if A² = I. 
• Upper Triangular Matrix: A square matrix in which all the elements below the diagonal are zero is known as the upper triangular matrix
• Lower Triangular Matrix: A square matrix in which all the elements above the diagonal are zero is known as the lower triangular matrix
• Singular Matrix: A square matrix is said to be a singular matrix if its determinant is zero i.e. |A|=0
• Nonsingular Matrix: A square matrix is said to be a non-singular matrix if its determinant is non-zero.

Determinant of a Matrix

The determinant of a matrix is a number associated with that square matrix. The determinant of a matrix can only be calculated for a square matrix. It is represented by |A|. The determinant of a matrix is calculated by adding the product of the elements of a matrix with their cofactors.

Minor of a Matrix

Minor of a matrix for an element is given by the determinant of a matrix obtained after deleting the row and column to which the particular element belongs to. Minor of Matrix is represented by Mij.

Inverse of a Matrix

A matrix is said to be an inverse of matrix ‘A’ if the matrix is raised to power -1 i.e. A-1. The inverse is only calculated for a square matrix whose determinant is non-zero. The formula for the inverse of a matrix is given as:

• A^-1 = adj(A)/det(A) = (1/|A|)(Adj A), where |A| should not be equal to zero, which means matrix A should be non-singular. 

Properties Inverse of Matrix

• (A^-1)^-1 = A 
• (AB)^-1 = B^-1A^-1 
• only a non-singular square matrix can have an inverse. 

Elementary Operation on Matrices

Elementary Operations on Matrices are performed to solve the linear equation and to find the inverse of a matrix. Elementary operations are between rows and between columns. There are three types of elementary operations performed for rows and columns. These operations are mentioned below:

The Elementary operations on rows include:

• Interchanging two rows
• Multiplying a row by a non-zero number
• Adding two rows

The Elementary operations on columns include:

• Interchanging two columns
• Multiplying a column by a non-zero number
• Adding two columns

Matrix Rank

• The rank of a matrix signifies the maximum count of linearly independent rows or columns within the matrix. It's always less than or equal to the total number of rows or columns present in the matrix. For a square matrix, linear independence of rows or columns implies non-singularity, meaning the determinant is not zero. Conversely, a zero matrix, devoid of linearly independent rows or columns, has a rank of zero.
• Calculation of matrix rank typically involves transforming the matrix into Row-Echelon Form. Through row echelonization, we aim to nullify all elements in a row using Elementary Row Operations. Subsequently, the count of rows containing at least one non-zero element post-operation determines the matrix rank, denoted as ρ(A).

Eigenvalues and Eigenvectors of Matrices

• Eigenvalues represent a set of scalars linked to linear equations in matrix form, sometimes referred to as characteristic roots of matrices. Eigenvectors, on the other hand, are vectors utilized to indicate direction at specific points, determined by the associated eigenvalues. Eigenvalues alter the magnitude of eigenvectors, which remain invariant under linear transformations.
• For a square matrix A of order 'n', another square matrix A – λI is constructed, where I denotes the Identity Matrix and λ represents the eigenvalue. This eigenvalue λ satisfies the equation Av = λv, where v signifies a non-zero vector.

Matrices Formulas

The basic formula for the matrices has been discussed below:

• A^-1 = adj(A)/|A|
• A(adj A) = (adj A)A = I, where I is an Identity Matrix
• |adj A| = |A|n-1 where n is the order of matrix A
• adj(adj A) = |A|n-2A where n is the order of the matrix
• |adj(adj A)| = |A|(n-1)^2
• adj(AB) = (adj B)(adj A)
• adj(Ap) = (adj A)p
• adj(kA) = kn-1(adj A) where k is any real number
• adj(I) = I
• adj 0 = 0
• If A is symmetric then adj(A) is also symmetric
• If A is a diagonal Matrix then adj(A) is also a diagonal matrix
• If A is a triangular matrix then adj(A) is also a triangular matrix
• If A is a singular Matrix then |adj A| = 0
• (AB)^-1 = B^-1A^-1