 Hi and welcome to the session. Today we will learn about minors and cofactors. First of all we will start with minors of an element aij of a determinant is the determinant obtained by deleting its ith row jth column in which the element aij lies and minor of an element aij is denoted by, for example, if in the given determinant we want to find out the minor of the element a32 then for that we will delete the third row and the second column in which the element lies and thus m32 that is the minor of the element a32 will be the determinant obtained by deleting the third row in the second column so it will be a11, a13, a21, a23. Now if the determinant is of order three then the minor of any element will be of order two that is one less than the order of the determinant that is one less than three. Now let's see what are cofactors. Cofactor of an element aij denoted by aij is defined as aij is equal to minus one to the power i plus j into mij where mij is the minor of the element aij so here the cofactor of the element a32 denoted as a32 will be given by minus one to the power three plus two into m32. Now we know how to find out the minors and cofactors of the elements of the given determinant so the value of this determinant will be if we expand it along r1 then it will be a11 into cofactor of the element a11 denoted by a11 plus a12 into the cofactor of the given element that is a12 plus element a13 into the cofactor of this element that is a13 which is equal to sum of the product of elements of r1 with their corresponding cofactors. Remember one thing that the sum of the product of elements of any row or column with the cofactors of any other row or column is zero. Now let's move on to our next topic that is a joint and inverse of a matrix. So first of all let's see what is a joint of a matrix the joint of a square matrix a given by aij of order n is defined as the transpose of the matrix given by aij of order n by n where aij is the cofactor of the element aij the joint of a matrix is denoted by a joint of a for example for the given matrix a a joint of a will be given by transpose of the matrix of the cofactors of the given elements which will be this one and its transpose will be this matrix. Now suppose we are given a two by two matrix a and we want to find a joint of a then there is a shortcut method for this what we will do is we will interchange these two elements so we will get a two two over here and a one one over here and for these two elements we will change the signs that is from plus to minus and from minus to plus so here we already have plus so this will be minus of a one two and minus of a two one so this will be the a joint of matrix a now let's see some important results first result is a b any given square matrix of order n then a into a joint of a is equal to a joint of a into a which is equal to determinant of a into i where i is the identity matrix of order n now before moving on to our next result we need to know some important properties that is i square matrix a is said to be singular if determinant of a is equal to zero also a square matrix a is said to be non-singular if determinant of a is not equal to zero now we'll move on to our next result that is result second this is states that if a non-singular matrices of the same order then a into b and b into a are also non-singular matrices of the same order now let's see the result the determinant of the product of matrices is equal to product of their respective determinants that is determinant of matrix a into b is equal to determinant of matrix a into determinant of matrix b where a and b are square matrices of the same order the fourth and the last result is a square matrix a is invertible and only if a is non-singular now let's take one example here we are given a matrix a and we need to find a inverse if it exists so let's find out determinant of a this will be equal to 14 which is not equal to zero this means matrix a is non-singular now by the result four we know that a square matrix a is invertible if and only if a is non-singular matrix so this means matrix a is invertible or a inverse exists also we have a inverse equal to 1 upon determinant of a into a joint of a so this will be equal to 1 upon 14 into a joint of a that is 3 minus 4 2 2 in the same way we can find out the inverse of a square matrix of order 3 with this we finished this session hope you must have enjoyed it goodbye take care and have a nice day