Question

If matrix$\mathit{B}$ is obtained by swapping two rows of a $\mathit{n}\mathbf{\times }\mathit{n}$matrix $\mathit{A}$, then the equation$\mathit{d}\mathit{e}\mathit{t}\mathit{B}\mathbf{=}\mathbf{-}\mathit{d}\mathit{e}\mathit{t}\mathit{A}$ must hold.

Short answer

Therefore, the given condition is true.

Step-by-step solution

Matrix Definition

Matrix is aset of numbers arranged in rows and columns so as to form a rectangulararray.

The numbers are called the elements, or entries, of the matrix.

If there are $m$ rows and $n$ columns, the matrix is said to be a “ $m$by $n$” matrix, written “ $m\times n$.”

To check whether the given condition is true or false

We can see that this also applies from the definition.

If $B$is obtained from by $A$switching the $i$-th and $j$-th row, and an added

$(-1{)}^{I\left(p\right)}\prod _{k=1}^n{a}_{k,p\left(k\right)}$

By calculating $\det A$, corresponds to a permutation $p\in S_n$, then the same added while calculating $\det B$ will correspond to the permutation where $p\left(i\right)$and $p\left(j\right)$switch places, so the addend will change its sign.

Thus,

$\det B=-\det A$