GATE CSE 2023 | Question: 8

Question

Let
\[
A=\left[\begin{array}{llll}
1 & 2 & 3 & 4 \\
4 & 1 & 2 & 3 \\
3 & 4 & 1 & 2 \\
2 & 3 & 4 & 1
\end{array}\right]
\]
and
\[
B=\left[\begin{array}{llll}
3 & 4 & 1 & 2 \\
4 & 1 & 2 & 3 \\
1 & 2 & 3 & 4 \\
2 & 3 & 4 & 1
\end{array}\right]
\]
Let $\operatorname{det}(A)$ and $\operatorname{det}(B)$ denote the determinants of the matrices $A$ and $B,$ respectively.

Which one of the options given below is $\text{TRUE}?$

• A. $\operatorname{det}(A)=\operatorname{det}(B)$
• B. $\operatorname{det}(B)=-\operatorname{det}(A)$
• C. $\operatorname{det}(A)=0$
• D. $\operatorname{det}(A B)=\operatorname{det}(A)+\operatorname{det}(B)$

Comments

The correct answer is Option : B

Answer 1

the elements of both the matrix $A,B$ are the same. in matrix $B$ $R_1,R_3$ are interchanged.

as we know the important property of any determinants: 

• if two rows (or 2 columns) of a determinant are interchanged the sign of the value of the determinant is changed.

here only one time rows is changed so the determinant should be multiplied by $(-1)$.

 $\therefore det (B)= -det(A)$

Option $(B)$ is correct.

Answer 2

Answer: B

In matrix $A$ after swapping row $1$ and row $3$ we will get matrix $B$. And when we swap two rows in a matrix it’s deteminant changes it’s sign so (B) is correct.

Answer 3

det(B) = – det(A)

 

Answer 4

Whenever we interchange a row or a column, a negative sign is introduced in the determinant value.

Here, in matrix B, if we perform an elementary operation, R1 ↔ R3, it becomes equal to Matrix A.

Hence, det(B) = – det(A)

Option (b) is correct.