even swaps means no change in sign. odd no of swaps means -1*sign of determinant

for a permutation matrix determinant is 1 if to reach that matrix the no of swaps taken to reach it from the identity matrix is even and -1 if odd

for 4th if we exchange the 2 identical rows the matrix doesn’t change, the determinant of original is D but due to row swap the determinant of new matrix is -D. but the fundamental matrix elements didn’t change means ⇒ D=-D ⇒D=0

for 5th property let their be a matrix with [r1,r2,r3…,rn] as the rows of the matrix let’s say i write

some row rj as rl + constantri. using property 3 we can write the original determinant as sum of 2 determinants one with rl in the place of rj and the other with lri inplace of rj (called say S)

for matrix S rj is lri from property 3 it can be written as ldet(S’) where rj is now ri

using property 4 det(S’)=0

hence rule 5 is proven and doing such an operation doesn’t change the determinant

using elimination remove all the upper triangular elements and remove out the diagonal elements what left is the identity matrix hence the result above

When A is singular there are dependent columns in A ⇒ using properties of determinant we can get a row of zeros ⇒ det (A)=0