Inverse of a matrix Questions

Consider the following in respect of the matrices (I) PQ is null matrix.

 (II) QP is an identity matrix of order 3.

 (III) PQ = QP.

Which of the above is/are correct?

What is the value of the determinant of the inverse of the matrix

The system of equations

 2x - 3y - 5 = 0, 15y - 10x + 50 = 0

In obtaining the solution of the system of equations x + y + z = 7, x + 2y + 3z = 16 and x + 3y + 4z = 22 by Cramer’s rule, the value of y is obtained by dividing D by D2, whereWhat is the value of the determinant D2?

Consider the following in respect of non- singular matrices A and B:

 I. (AB) ^{- 1} = A^–1B^–1

II. (BA)(AB)^–1 = I, where I is the identity matrix

III. (AB)^T = A^T B^T

How many of the above are correct?

The value of the determinant is equal to

If a, b, c are the sides of a triangle ABC and p is the perimeter of the triangle, then what is equal to?

For the following two (02) items:

 Let A

What is the value of the determinant of the matrix A4?

For the following two (02) items:

 Let A

What is [adj A]^–1 equal to?

Consider the following statements:

Statement-I: If X is an n × n matrix, then det (mX) = mn det(X), where m is a scalar.

Statement-II: If Y is a matrix obtained from X by multiplying any row or column by a scalar m, then det(Y) = mdet (X). Which one of the following is correct in re- spect of the above statements?