 Hi and welcome to the session. Today we will learn about determinant. To every square matrix A of order n we can associate a number real or complex called determinant of the square matrix A. So if m is the set of square matrices and k is the set of numbers real or complex f is a function from m to k defined as f of A is equal to k, A belongs to m, A belongs to k. Then f of A is called the determinant of A and it is denoted by this symbol or determinant of A on this symbol. So if we are given a matrix A then determinant of A is written as determinant of A square matrix of order is called determinant of order n also only square matrices of determinants. Now let us see the determinant of a matrix of order 1. We are given a matrix B of order 1. So there will be just one element say A in the matrix then the determinant of matrix B will be the element A itself. So A is the determinant of matrix B. Now let us move on to determinant of a matrix of order pose. We are given a matrix A of order 2. Then the determinant of matrix A written as this or this will be given as this one. So this will be the determinant of order 2 and this will be equal to here we will cross multiply the elements like this. So this determinant will be equal to A11 into A22 minus A21 into A12. So this is how we find the determinant of order 2. Let us take one example for this. Here we are given a matrix A and we need to find the determinant of A. So this will be equal to now we will cross multiply the elements. So this will be 1 into minus 4 into minus 2 and this will give us 10. So determinant of A is equal to 10. Now let us move on to our next topic determinant of a matrix of order 3. We need to find out determinant of A which will be the determinant of order 3. Now we can find determinant of order 3 in 6 ways that is expanding the determinant corresponding to any of the 3 rows or any of the 3 columns. So here first of all we will learn how to find out the determinant by expansion along first row that is R1. So let us find out the determinant of A. Now first of all we will multiply the first element of R1 that is A11 by minus 1 to the power 1 plus 1. Now here we have added the suffixes of the element A11 that is 1 and 1. Now the element A11 lies in the first row also in the first column. So we will delete the first row and the first column. Now we will multiply the determinant obtained by the remaining elements that is a determinant of order 2. This will be A22, A23, A32, A33. Now in the same manner we will take the second element of R1 that is A12 and we will add in this A12 and we will multiply A12 by minus 1 to the power 1 plus 2 that is the suffixes of the element A12. Now the element A12 lies in the first row and second column. So let us delete these two. Now again we will get a determinant of order 2 of the remaining elements. So we will multiply it with that A21, A23, A31, A33. Now we will take the third element of R1 that is A13. So here let us add A13 and we will multiply it by minus 1 to the power 1 plus 3 that is the suffixes of the element A13. Again same way A13 lies in the first row and third column. So we will delete these two and we will get a determinant of order 2 of the remaining elements. So we will multiply this determinant by this that is A21, A22, A31, A32. So this is the determinant of A and now this will be equal to minus 1 to the power 1 plus 1 means 1, 1 into A11 is A11 into. Now we know how to find out the determinant of order 2. So this will be A22 into A33 minus A32 into A23. Now minus 1 to the power 1 plus 2 that is minus 1 into A12. So this will be minus A12 into A21, A33 minus A31 into A23. Now plus A13 into A21, A32 minus A31 into A22. Similarly we can find out the determinant by expanding this determinant along R2, R3 or C1, C2 or C3 and there is one important point that is expanding a determinant along any row or column gives same value. Now for the easier calculations we shall expand the determinant along that row or column which contains maximum number of zeros. So let's take one example for this. Here we need to find out determinant of A. Now as we can see there are two zeros in second row. So we will find out the determinant by expansion along second row that is R2. Here first of all we will take the first element of R2. So we will multiply 0 by minus 1 to the power 2 plus 1 as this is the A21 element of the determinant A. Now this element lies in the second row and first column so we will delete them and we will multiply this with the determinant of the remaining elements. So this will be minus 1 into 0 minus minus 5 into minus 2 plus. Now let's take the second element of R2 that is 0 again. So we will multiply 0 by minus 1 to the power 2 plus 2. Now let's delete the second row and second column and let's multiply it with the determinant of order 2 and we will get 3 into 0 minus 3 into minus 2 plus. Now let's take the third element of R2 that is minus 1. So we will multiply minus 1 with minus 1 to the power 2 plus 3 and we will delete the second row and third column as minus 1 lies in the second row and third column. So let's multiply this with the determinant of order 2 of the remaining elements. This will be 3 into minus 5 minus 3 into minus 1. Now this will be equal to 0 plus 0 plus 1 into minus 15 plus 3 that is equal to minus 12. So determinant of A is minus 12. So in this session we have learned to find out the determinants of order 1, 2 and 3. With this we finish this session. Hope you must have understood all the concepts. Goodbye, take care and have a nice day.