Linear Algebra Chapter 3.2

A row replacement operation does not affect the determinant of the matrix

True. Thm 3 Part A

The determinant of A is the product of the pivots in any echelon form U of A, multiplied by (-1)^r, where r is the number of row interchanges made during row reduction from A to U


If the columns of A are linearly dependent, then det A = 0

True. If the columns of A are Linearly Dependent, then by the Invertible matrix theorem the matrix A formed by the columns of A is not invertible and thus by Thm. 4, det A = 0.

det( A + B) = det A + det B


If three row interchanges are made in succession, then the new determinant equals the old determinant

False (-1)^3 = -1 therefore, the determinant is not the same

The determinant of A is the product of the diagonal entries in A

False, must be in echelon form

If det A is zero, then two rows or two columns are the same, or a row or a column is zero

False. Not necessarily. There are other conditions where the determinant can be zero. But yes, if 2 rows or coluns or any row or column is zero. determinant is also zero

det A^-1 = (-1)det A

False. Det(Inverse A) = 1/detA but not (-1) detA

if B is obtained by adding a multiple of one row to another

Det B = Det A

if B is obtained from A by interchanging two rows

det B = - det A

If B is obtained from A by multiplying one row/column by k

det B = k det A

A is invertible if and only if

the determinant is not zero



detAB =

detA * detB

detA = -3detB = 4det 5A =


detA = -3detB = 4detB^T


detA = -3detB = 4det A^-1


detA = -3detB = 4det A^3


det A = -3det B = -1det AB


det A = -3det B = -1detA^T B A


det A = -3det B = -1det B^-1 A B