Question

How do you simplify determinants?

Answer 1

Hint: A determinant refers to a function that can start from the set of zero square matrices to the set of real numbers. Every square matrix has their determinant. Only the matrices which are square matrices can have determinant, matrices other than square matrices do not have determinant.
Determinant of any square matrix can be represented as det (A) or \[\left| A \right|\] or \[\Delta \].
Suppose a square matrix;\[A=\left[ \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right]\]then its determinant is,\[\left| A \right|=\left| \begin{matrix}
   a & b \\
   c & d \\
\end{matrix} \right|\]

Complete step by step solution:
We can use the following method to simplify the given determinant;
Let \[\Delta =\left| \begin{matrix}
   {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\
   {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\
   {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\
\end{matrix} \right|\]
Thus,
\[\Delta ={{a}_{1}}\left| \begin{matrix}
   {{b}_{2}} & {{b}_{3}} \\
   {{c}_{2}} & {{c}_{3}} \\
\end{matrix} \right|-{{a}_{2}}\left| \begin{matrix}
   {{b}_{1}} & {{b}_{3}} \\
   {{c}_{1}} & {{c}_{3}} \\
\end{matrix} \right|+{{a}_{3}}\left| \begin{matrix}
   {{b}_{1}} & {{b}_{2}} \\
   {{c}_{1}} & {{c}_{2}} \\
\end{matrix} \right|\]
\[\Delta ={{a}_{1}}\left( {{b}_{2}}{{c}_{3}}-{{b}_{3}}{{c}_{2}} \right)-{{a}_{2}}\left( {{b}_{1}}{{c}_{3}}-{{b}_{3}}{{c}_{1}} \right)+{{a}_{3}}\left( {{b}_{1}}{{c}_{2}}-{{b}_{2}}{{c}_{1}} \right)\]
Some properties of determinants are,
i.If any two rows or any two columns of a given determinant are being interchanged, then the sign of the determinant changes.
ii.In the given determinant, if two rows or two columns are identical then the value of the determinant is equal to zero.
iii.If each element of a row or a column is zero then the value of the determinant is also zero.

Note: Students need to remember to simplify the given determinant is that; first take a common factor to taking away the constant out from any row or from any column, second we can add any multiple of one row to the another row and same with columns and the third is we can interchange any two rows or column.