Which of the following statements are true Which are false a

Which of the following statements are true? Which are false? (a) If A is a 3 times 3-matrix with a zero determinant, then one row of A must be a multiple of some other row. (b) Even if any two rows of a square matrix are equal, the determinant of that matrix may be non-zero. (c) If any two columns of a square matrix are equal then the determinant of that matrix is zero. (d) For any pair of n x n-matrices, A and B, we always have det(A + B) = det(A) + det(B). (e) Let A be an 3 times 3-matrix. Then det(7A) = 7^3det(A). (f) If A^-1 exists, then det(A^-1) = det (A).

Solution

a)

False

One can have the first row being a sum of second and third row.

Since determinant is invariant under elementary row operations

So we can perform the row operation: R_1=R_1-R_2-R_3

which transform the first row elements to 0 and ehnce determinant is 0

b)

False

If two rows are equal one of them can be eliminated by an elementary row operation hence determinant would be 0

c)

True

d)

False,

Let, A=I ie identity matrix

B=A

det(A+B)=det(2A)=det(2I)=4

det(A)+det(B)=2det(A)=2*1=2

e)

True. In 7A each element gets multiplied by 7 and in a determinant formula each term in summation is product of three entries of A. Hence each term in the sum gets multiplied by 7^3 and hence determinant is multiplied by 7^3

f)

False

det(AA^{-1})=det(I)=1=det(A)det(A^{-1})

This is the correct relationship