Question

Consider the following statements:

I. If any two rows or columns of a determinant are identical, then the value of the determinant is zero.

II. If the corresponding rows and columns of a determinant are interchanged, then the value of the determinant does not change.

III. If any two rows (or columns) of a determinant are interchanged, then the value of the determinant in sign.

Which of these is correct?

• A. $IandII$
• B. $IIandIII$
• C. $I,II,andIII$
• D. $IandIII$

Answer 1

The correct option is C

$I,II,andIII$

Properties of determinant :

I: A determinant can be expanded along a row or a column. If two rows are identical, then we can subtract one row from another. If one row of a determinant is zero, the value of the determinant is zero, subtracting one row from an identical row makes the values of that row zero giving zero to expanding determinant. Similarly, if two columns are identical we can interchange rows and columns and subtract one row from an identical row making the values of row zeros on expanding the determinant value will be zero.

II: For any matrix,$m\times n$​ value of the determinant is$a_{\left|x\right|}\times$co-factor of$a_{1\times 1}+...+a_{1\times n}$​ cofactor of$a_{1\times n}$ ​ i.e. value of the determinant is the summation of the product of an element and its respective co-factor. So, the matrix can be expanded along any column or any row.

III: Say a matrix has $m$ rows and $n$ columns. In a matrix, if rows and columns are interchanged values of the matrix change in sign.

Hence, the correct option is C.