Question

Are the following statements true or false? 1. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix. 2. If det(A) is zero, then two rows or two columns are the same, or a row or a column is zero. 3. The determinant of A is the product of the diagonal entries in A. 4. det(AT)=(−1)det(A).

Answer 1

Answer:

1. True.
2. True.
3. False.
4. False.

Step-by-step explanation:

1. If two row interchanges are made in sucession, then the determinant of the new matrix is equal to the determinant of the original matrix. - True.  

If you interchange two rows in sucession, then the determinant of the obtained matrix is equal to determinant of the original matrix, so this statement is true.

2. If det(A) is zero, then two rows or two columns are the same, or a row                       or a column is zero. - True.

A determinant is equal to 0 if

• two rows or columns are the same or in a proportion
• any row or column consist of zeros only.

3. The determinant of A is the product of the diagonal entries in A. - False.

We determine the value of a determinant by choosing a single row or a column (usually, we choose the ones which have zeros, to make the calculation faster and easier). Then, we cross the row and column of the first element and find the determinant of the smaller matrix obtained. Then, we multiply it by the chosen element and its sign  determined by $(-1)^{i+j}$, where $i, j$ are its row and column. We repeat the process for each element from the chosen column or row.

Therefore, the value of the determinant is not equal to the product of the diagonal entries.

4.  $\det (A^T) = (-1) \det (A)$ - False.

Transposing a matrix does not change its determinant.