Question

A and B are n*n matrices. Check the true statements below. Please provide explanations. Thanks! -The determinants of A is the product of the diagonal entries in A. -If (lambda)+5 is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A. -(detA)(detB) = detAB -An elementary row operation on A does not change the determinant. -If(lambda-r)^k is a root of the characteristic polynomial of A, then r is an eigenvalue of A with algebraic multiplicity k. -If A is 3*3, with columns a1,a2,a3, then detA equals the volume of the parallelpiped determined by vectors a1,a2,a3. -If one multiple of one row of A is added to another row, the eigenvalues do not change. -detA(transpose) = (-1)detA

Answer 1

For a 2*2 matrix the determinant is left diagonal minus right diagonal. For 3*3 matrix we follow the Chess board rules Plus minus plus first row. Second row is minus plus minus while Third is same with first multiply along diagonal with elimination row column step-by-step.

Explanation: