Matrices and Determinants
Matrices and Determinants PDF Notes, Important Questions and Synopsis
SYNOPSIS
 Determinants
Every square matrix can be associated to a number called a Determinant.
For A → Square matrix; A or det A or D → denotes the determinant of A

Submatrix
A matrix obtained after deleting some rows or columns is called a submatrix. 
Minor & Cofactor
A minor is the determinant of the submatrix obained by deleting the i^{th} row and j^{th} column.
It is denoted by M_{ij}.
A cofactor is denoted by C_{ij}, and it is given by C_{ij} = (1)^{i+j} M_{ij}. 
Finding Determinant
A matrix should be a square matrix of order greater than 1, A = [a_{ij}]_{nxn}.
A determinant of a matrix A is defined as the sum of the products of elements of any one row (or column) with corresponding cofact
A= a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} (Using first row)ors. 
Properties of a Determinant
 The value of a determinant remains the same if the rows and columns are interchanged.
 The value of a determinant changes in sign only if any two rows (or columns) of a determinant are interchanged.
 If a determinant has any two rows (or columns) identical, then the value of the determinant is zero.
 If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.
 If each element of any row (or column) can be expressed as a sum of two terms, then the determinant can be expressed as the sum of two determinants.
 The value of a determinant does not change by adding to the elements of any row (or columns) the same multiples of the corresponding elements of any other row.
Note: While applying this property, at least one row or column must remain unchanged.

Cramer’s Rule (Determinant)
Consider three simultaneous linear equations:
a_{1}x + b_{1}y + c_{1}z = d_{1} , a_{2}x + b_{2}y + c_{2}z = d_{2} and a_{3}x + b_{3}y + c_{3}z = d_{3The solution of the above system of linear equations is :}
A matrix is a rectangular array/arrangement of numbers along rows and columns.
Note: A matrix A = [aij], where aij is an element of the i^{th} row and j^{th} column.
Row number and column number are the same for diagonal elements.
 Classification of Matrices
 Row Matrix A = [3 6 8]
Matrix having only OR
one row. A = [1 2 6 0]
 Column Matrix
Matrix having only one column.
 Square Matrix
Matrix having same number of rows and columns.
 Zero (Null) Matrix
Matrix having all elements equal to zero.
Matrix having all elements equal to zero.
 Upper Triangular Matrix
All entries below the main diagonal are zero.
All entries below the main diagonal are zero.
 Lower Triangular Matrix
All entries above the main diagonal are zero.
 Diagonal Matrix
All entries above and below the diagonal are zero.
All entries above and below the diagonal are zero.
 Identity (Unit) Matrix
All diagonal entries are one and the rest are zero.
All diagonal entries are one and the rest are zero.
 Row Matrix A = [3 6 8]
 Operations on Matrices
 Addition of Matrices
Order of the matrices must be same.
 Subtraction of Matrices
Order of the matrices must be same.
 Equality of matrices
Matrices having same order with each of their elements equal.
 Transpose of a Matrix
Matrix obtained after turning rows into columns and vice versa; denoted by A^{T}.
Matrix obtained after turning rows into columns and vice versa; denoted by A^{T}.
 Addition of Matrices
 Multiplication of Matrices
 Multiplication of a matrix with a scalar
Each element of the matrix is multiplied by the scalar.
 Multiplication of two matrices
Let A and B be two matrices, then A
Let A and B be two matrices, then A ´ B is possible only if ‘No. of columns of 1^{st} matrix = No. of rows of 2^{nd} matrix’.
Process of Multiplication of two Matrices
 Multiplication of a matrix with a scalar
 Inverse of a Matrix
Inverse of a matrix A is denoted by A^{1} such that (matrix) (inverse of matrix) = I
i.e. A × A^{1} = I or A^{1} × A = I. But A × A^{1} ≠ A^{1} ´ A.
Steps of finding inverse of a matrix:Step I
Check whether matrix A is singular or nonsingular, i.e.
A = 0 Þ Singular
A ≠ 0 Þ NonsingularStep II
If matrix A is nonsingular, then find the value of determinant of A and also find the adjoint matrix A.
Step III
Use the formula
Note: if A is nonsingular.
If A = diag (a_{11}, a_{22}, …., a_{nn}), then  Type of Square matrices
 Nilpotent Matrix
If for some least +ve integer p, then A^{p} =0 is a nilpotent matrix.
Idempotent Matrix
If A^{2}= A then A is an idempotent matrix.
Symmetric Matrix
If A^{T} = A, then A is a symmetric matrix.Skewsymmetric Matrix
If A^{T} = A, then A is a skewsymmetric matrix.
Also, all the diagonal elements are zero.Involutory Matrix
If A^{2} = I, then A is an involutory matrix.Unitary Matrix
If A’(A’)^{T} = I, where A’ is the complex conjugate of A, then A is a unitary matrix.Orthogonal Matrix
If A is a square matrix such that A^{T}A = I = A^{T}A or A^{T} = A^{1}, then A is an orthogonal matrix.  Nilpotent Matrix
 For any square matrix A, A + A^{T} is symmetric and A  A^{T} is skewsymmetric.
 Every square matrix can be expressed as a sum of symmetric and skewsymmetric matrices.
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