Mark each of the following statements true or false Briefly

Mark each of the following statements true or false. Briefly (!) justify your answer. A. An n times n matrix with two identical rows is never invertible. B. An n times n matrix with a zero row or column can be invertible. C. An n times n matrix where one row or column is a multiple of another row or column can never be invertible. D. A 3 times 3 matrix with all zeros on the diagonal cannot be invertible. D. If A is an n X n matrix such that A^2 = 0, then I_n + A is invertible.

Solution

a}True; a square matrix with two identical rows/columns has determinant zero. & an invertible matrix has non-zero determinant.
b}False;a square matrix a zero row or column has determinant zero. & an invertible matrix has non-zero determinant.
c}True;a square matrix,where one row or column is a multiple of another row has determinant zero. & an invertible matrix has non-zero determinant.
d}False;a square matrix with all zeroes on the diagonal has non-zero determinant . & an invertible matrix has non-zero determinant.