Engineering Mathematics

Relation Between \(\det(A)\) and \(\det(B)\)

Question

Let

A= \begin{bmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3\\ 3 & 4 & 1 & 2\\ 2 & 3 & 4 & 1 \end{bmatrix}

and

B= \begin{bmatrix} 3 & 4 & 1 & 2\\ 4 & 1 & 2 & 3\\ 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{bmatrix}

Let \(\det(A)\) and \(\det(B)\) denote the determinants of matrices \(A\) and \(B\), respectively.

Which one of the following options is true?

A. \(\det(A)=\det(B)\)

B. \(\det(B)=-\det(A)\)

C. \(\det(A)=0\)

D. \(\det(AB)=\det(A)+\det(B)\)

Given

Matrix \(A\) is:

A= \begin{bmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3\\ 3 & 4 & 1 & 2\\ 2 & 3 & 4 & 1 \end{bmatrix}

Matrix \(B\) is:

B= \begin{bmatrix} 3 & 4 & 1 & 2\\ 4 & 1 & 2 & 3\\ 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{bmatrix}

We need to determine the relationship between \(\det(A)\) and \(\det(B)\).

Concept Used

Determinant property for interchange of rows

When any two rows of a determinant are interchanged, the sign of the determinant changes.

\boxed{ R_i\leftrightarrow R_j \quad\Longrightarrow\quad \det(B)=-\det(A) }

For example, if

A= \begin{bmatrix} R_1\\ R_2\\ R_3\\ R_4 \end{bmatrix}

and we interchange \(R_1\) and \(R_3\), then

B= \begin{bmatrix} R_3\\ R_2\\ R_1\\ R_4 \end{bmatrix}

and therefore:

\boxed{ \det(B)=-\det(A) }

Solution

Step 1: Write the rows of \(A\)

The matrix \(A\) is:

A= \begin{bmatrix} 1 & 2 & 3 & 4\\ 4 & 1 & 2 & 3\\ 3 & 4 & 1 & 2\\ 2 & 3 & 4 & 1 \end{bmatrix}

Denote its rows by:

R_1= \begin{bmatrix} 1&2&3&4 \end{bmatrix} \] \[ R_2= \begin{bmatrix} 4&1&2&3 \end{bmatrix} \] \[ R_3= \begin{bmatrix} 3&4&1&2 \end{bmatrix} \] \[ R_4= \begin{bmatrix} 2&3&4&1 \end{bmatrix}

Step 2: Compare matrix \(B\) with matrix \(A\)

Matrix \(B\) is:

B= \begin{bmatrix} 3 & 4 & 1 & 2\\ 4 & 1 & 2 & 3\\ 1 & 2 & 3 & 4\\ 2 & 3 & 4 & 1 \end{bmatrix}

Now compare each row of \(B\) with the rows of \(A\).

The first row of \(B\) is:

\begin{bmatrix} 3&4&1&2 \end{bmatrix} = R_3

The second row remains:

\begin{bmatrix} 4&1&2&3 \end{bmatrix} = R_2

The third row of \(B\) is:

\begin{bmatrix} 1&2&3&4 \end{bmatrix} = R_1

The fourth row remains:

\begin{bmatrix} 2&3&4&1 \end{bmatrix} = R_4

Therefore, \(B\) is obtained from \(A\) by:

\boxed{ R_1\leftrightarrow R_3 }

Step 3: Apply the determinant row-interchange property

We know the fundamental determinant property:

\boxed{ \text{Interchanging any two rows changes the sign of the determinant} }

Here, exactly one row interchange has been performed:

R_1\leftrightarrow R_3

Therefore:

\det(B) = -\det(A)

Hence:

\boxed{ \det(B)=-\det(A) }

Step 4: Check the options

We obtained:

\boxed{ \det(B)=-\det(A) }

Comparing this with the options:

\boxed{ \text{Option B} }

Therefore, the correct answer is Option B.

Why don't we calculate the determinants?

A common mistake is to calculate the \(4\times4\) determinant of \(A\) and then calculate the \(4\times4\) determinant of \(B\).

That is completely unnecessary.

The question asks for the relationship between the two determinants, not their individual numerical values.

Since \(B\) is obtained from \(A\) simply by interchanging two rows, the determinant property gives the answer immediately.

A \xrightarrow{R_1\leftrightarrow R_3} B \] \[ \therefore \boxed{ \det(B)=-\det(A) }

Row-by-Row Verification

Starting with \(A\):

A= \begin{bmatrix} R_1\\ R_2\\ R_3\\ R_4 \end{bmatrix}

Interchange \(R_1\) and \(R_3\):

\begin{bmatrix} R_1\\ R_2\\ R_3\\ R_4 \end{bmatrix} \xrightarrow{R_1\leftrightarrow R_3} \begin{bmatrix} R_3\\ R_2\\ R_1\\ R_4 \end{bmatrix}

Substituting the actual rows:

\begin{bmatrix} 1&2&3&4\\ 4&1&2&3\\ 3&4&1&2\\ 2&3&4&1 \end{bmatrix} \xrightarrow{R_1\leftrightarrow R_3} \begin{bmatrix} 3&4&1&2\\ 4&1&2&3\\ 1&2&3&4\\ 2&3&4&1 \end{bmatrix}

The resulting matrix is exactly \(B\).

\boxed{ A \xrightarrow{R_1\leftrightarrow R_3} B }

Final Answer

\boxed{ \det(B)=-\det(A) }

Correct option: B

Formulas & Concepts to Remember

1. Interchange of two rows

If two rows of a determinant are interchanged:

\boxed{ R_i\leftrightarrow R_j \Rightarrow \det(B)=-\det(A) }

2. Two row interchanges

If two row interchanges are performed:

(-1)^2=+1 \] \[ \boxed{ \det(B)=\det(A) }

3. \(k\) row interchanges

\boxed{ \det(B)=(-1)^k\det(A) }

4. Determinant of a product

\boxed{ \det(AB)=\det(A)\det(B) }

GATE Tip

This is a classic property-based question.

Do not calculate a \(4\times4\) determinant unless the question actually asks for its value.

Whenever you see two matrices \(A\) and \(B\), compare their rows and columns first.

Here:

B = A \text{ with }R_1\text{ and }R_3 \text{ interchanged}

One interchange means one sign change:

\boxed{ \det(B)=-\det(A) }

Exam time: roughly 10–20 seconds if you recognize the row interchange immediately.

← Back to Engineering Mathematics

Determinant of Matrices — GATE CS 2023