Matrices and Determinants

SYNOPSIS

1. 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
   
    

2. Sub-matrix
   A matrix obtained after deleting some rows or columns is called a sub-matrix.

3. Minor & Cofactor
   A minor is the determinant of the sub-matrix 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.

4. 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|= a₁₁C₁₁ + a₁₂C₁₂ + a₁₃C₁₃ (Using first row)ors.
   
   

5. Properties of a Determinant
   

   1. The value of a determinant remains the same if the rows and columns are interchanged.
   2. The value of a determinant changes in sign only if any two rows (or columns) of a determinant are interchanged.
      
   3. If a determinant has any two rows (or columns) identical, then the value of the determinant is zero.
      
      
   4. If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.
      
      
   5. 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.
      
      
   6. 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.

   
   

6. Cramer’s Rule (Determinant)
   Consider three simultaneous linear equations:
   a₁x + b₁y + c₁z = d₁ ,  a₂x + b₂y + c₂z = d₂  and  a₃x + b₃y + c₃z = d_{3
   The 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.

1. 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.

2. 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 AT. Matrix obtained after turning rows into columns and vice versa; denoted by AT.

3. 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 1st matrix = No. of rows of 2nd matrix’.
   Process of Multiplication of two Matrices

4. 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 non-singular, i.e. |A| = 0 Þ Singular |A|   ≠ 0 Þ Non-singular

   Step II If matrix A is non-singular, then find the value of determinant of A and also find the adjoint matrix A.

   Step III Use the formula

   
   
   Note: if A is non-singular.
               If A = diag (a₁₁, a₂₂, …., a_nn), then 
5. Type of Square matrices
   
   

   Nilpotent Matrix If for some least +ve integer p, then Ap =0 is a nilpotent matrix.	Idempotent Matrix  If A2= A then A is an idempotent matrix.
   Symmetric Matrix    If AT = A, then A is a symmetric matrix.	Skew-symmetric Matrix  If AT = -A, then A is a skew-symmetric matrix.  Also, all the diagonal elements are zero.
   Involutory Matrix   If A2 = 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 ATA = I = ATA or AT = A-1, then A is an orthogonal matrix.

   
   
6. For any square matrix A, A + A^T is symmetric and A - A^T is skew-symmetric.
7. Every square matrix can be expressed as a sum of symmetric and skew-symmetric matrices.