GATE Overflow Test Series | Linear Algebra | Test 1 | Question: 6

Question

Which of the following is/are correct? (Mark all the appropriate choices)

• A. The value of a determinant remains unaltered, if the rows and columns are interchanged.
• B. If two rows (or two columns) of a determinant are interchanged, the sign of the value of the determinant changes.
• C. If two rows (or columns) of a determinant are identical, the value of the determinant is zero.
• D. If the elements of any row (or column) of a matrix are multiplied by a constant $k,$ the determinant also gets multiplied by $k.$

Answer 1

For option $(A):$ The determinant of a matrix remains unaltered if its rows are changed with columns. That is, $\mid A \mid  = \mid A^{T} \mid .$

Since the row-wise expansion is same as the column-wise expansion, the result holds good.

For option $(B):$ If two rows or columns are interchanged, then the value of determinant changes its sign.

For example,
Let $\mid A \mid = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad  - bc$

Interchanging rows

$\mid B \mid = \begin{vmatrix} c & d \\ a & b \end{vmatrix} = bc - ad = -(ad  - bc)$

Interchanging columns

$\mid C \mid = \begin{vmatrix} b & a \\ d & c \end{vmatrix} = bc - ad = -(ad - bc)$

For option $(C):$ Let the determinant of a matrix be $x.$ If we interchange the two identical rows of this matrix, then by the  property in option B, the determinant of the new matrix is $-x,$ but overall the matrix will be same as we have interchanged only the two identical rows.
So, $x = -x, \implies x = 0.$

Hence, the determinant is zero.

Option $(D)$ is also TRUE. Some examples:

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$

$\implies \mid A \mid = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$

Now, $\mid B \mid  = \begin{vmatrix} ka & kb \\ c & d \end{vmatrix} = k ad - k bc = k(ad - bc)$

$\text{(OR)}$

Now, $\mid C \mid  = \begin{vmatrix} ka & b \\ kc & d \end{vmatrix} = k ad - k bc = k(ad - bc)$

So, the correct answer is $A; B; C;D.$

Ref: https://www.math.fsu.edu/~fusaro/EngMath/Ch6/SPDR.html