CHAPTER 2 MATRICES Determinant Inverse

DETERMINANT : 2X2 Definition 2.19 Determinant of matrix A is defined by det(A) or |A| Definition 2.19 If is a 2x2 matrix, then the determinant is given by

INVERSE : 2X2 Definition If , then is the inverse of A where Theorem 1 Matrix A in invertible if and only if If , then A doesn’t have an inverse.

INVERSE : 2X2 Example 1 Find the inverse for the given matrix:

INVERSE : 2X2 Example 1 Find the inverse for each matrix:

INVERSE : 2X2 Example 2 If , show that is the inverse of A.

DETERMINANT : 3x3 Definition 2.20 Given is a 3x3 matrix, then the determinant is given by:

DETERMINANT : 3x3 (minors & cofactors) Definition 2.23 If Cij is cofactor of matrix A, then det(A) can be obtained by: Expanding along the ith row: Expanding along the jth column:

DETERMINANT : 3x3 (minors & cofactors) Definition 2.22 Cofactor of aij : For 3x3 matrix : We can conclude that : Even = 1, Odd = -1

DETERMINANT : 3x3 (minors & cofactors) Definition 2.21 Let n ≥ 2 and A = [ aij ]nxn . Matrices (n -1)x (n -1) submatrix of A is obtained by deleting the ith row and jth column of A, denoted by Mij . Minor of

DETERMINANT : 3x3 (minors & cofactors) Therefore : Cofactor of aij =

DETERMINANT : 3x3 (minors & cofactors) Exercise 1: Find determinant of A:

DETERMINANT : 3x3 (minors & cofactors) Answer 1 Minor= Cofactor =

DETERMINANT : 3x3 (minors & cofactors) Exercise 2: Find determinant of A: ANSWER:

DETERMINANT : PROPERTIES Suppose A is nxn matrix and k is a scalar. Suppose the matrix B is obtained by multiplying a single row or column of A by k. Then det(B) = k det(A) If matrix A is multiplied by k, that is every element in the matrix is multiplied by k, then det(kA) = kn det(A) If B is obtained from A by interchanging 2 rows or 2 columns, then det(B) = - det(A)

DETERMINANT : PROPERTIES Adding or subtraction a multiple of one row(column) to the other row(column) leaves the determinant unchanged If A and B are 2 square matrices such that AB exists, then, det(AB) = det(A) det(B) If 2 rows or 2 columns of a matrix are equal, the determinant of the matrix is zero.

ADJOINT Definition 2.24 Let A is an nxn matrix, then the transpose of the matrix of cofactors A is called the matrix adjoint to A.

ADJOINT Exercise 3: Find the inverse of A:

ADJOINT

ADJOINT Answer 3: Cofactor = adj(A) =

ADJOINT Exercise 4: Find inverse of B:

ADJOINT Answer 4: Cofactor = adj(B) =

INVERSE Definition 2.25 If A is a square matrix of order n and if there exists a matrix A-1 such that then A-1 is called the inverse of A.

ELEMENTARY ROW OPERATION INVERSE FOR 3X3 COFACTOR METHOD ELEMENTARY ROW OPERATION (ERO)

Inverse using Cofactor Method Theorem 2 If A is nxn matrix, |A|≠0, then A-1 is defined by:

INVERSE Exercise 7: Find the inverse of each matrix using the Cofactor Method:

INVERSE Answer 7: Solution for (a): Step 1: find cofactor of A Step 2: find det(A)

INVERSE Step 3: find adj(A) Step 4: find the inverse of A

Inverse using Elementary Row Operations (ERO) Theorem 3 Let A and I both be nxn matrices, the augmented matrix may be reduced to by using elementary row operation (ERO)

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (i) : interchange the elements between ith row and jth row Example

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (ii) : multiply ith row by a nonzero scalar, k Example NEW R1

Inverse using Elementary Row Operations (ERO) Characteristics of ERO (iii) : add or subtract ith row to a constant multiple jth row by a nonzero scalar, k Example NEW R1

Inverse using Elementary Row Operations (ERO) Exercise 8: Find the inverse of each matrix using the Elementary Row Operations (ERO)