Question

How do you multiply determinants?

Answer 1

Hint: In this problem, we can see how to multiply the determinants. We should know that determinants can be multiplied together only if they are of the same order. And also, the process of interchanging the rows and columns will not affect the value of the determinant. So, we can multiply determinants in various ways. We can now see the following procedures for multiplication of determinants are Row by row multiplication rule, Column by column multiplication rule, Row by column multiplication rule, Column by row multiplication rule.

Complete step by step solution:
We know that we can multiply determinants if they are of the same order. We also know that the process of interchanging the rows and columns will not affect the value of the determinant. Now, we can multiply determinants in various ways.
Let us consider two determinants
\[\begin{align}
  & A=\left| \begin{matrix}
   a & b & c \\
   d & e & f \\
   g & h & i \\
\end{matrix} \right| \\
 & B=\left| \begin{matrix}
   p & q & r \\
   s & t & u \\
   v & w & x \\
\end{matrix} \right| \\
\end{align}\]
If we multiply A and B using point (i),
\[A\times B=\left| \begin{matrix}
   {{R}_{1}}R{{'}_{1}} & {{R}_{1}}R{{'}_{2}} & {{R}_{1}}R{{'}_{3}} \\
   {{R}_{2}}R{{'}_{1}} & {{R}_{2}}R{{'}_{2}} & {{R}_{2}}R{{'}_{3}} \\
   {{R}_{3}}R{{'}_{1}} & {{R}_{3}}R{{'}_{2}} & {{R}_{3}}R{{'}_{3}} \\
\end{matrix} \right|\]
Where, \[{{R}_{i}}\] is row of first determinant i.e. A
\[R{{'}_{j}}\] is row of second determinant i.e. B
(j, i <=3, because order of determinant is 3)
Similarly, if A and B is multiplied using point (ii), we have
\[A\times B=\left| \begin{matrix}
   {{C}_{1}}C{{'}_{1}} & {{C}_{2}}C{{'}_{1}} & {{C}_{3}}C{{'}_{1}} \\
   {{C}_{1}}C{{'}_{2}} & {{C}_{2}}C{{'}_{2}} & {{C}_{3}}C{{'}_{2}} \\
   {{C}_{1}}C{{'}_{3}} & {{C}_{2}}C{{'}_{3}} & {{C}_{3}}C{{'}_{3}} \\
\end{matrix} \right|\]
Where, \[{{C}_{i}}\]is column of first determinant i.e. A
\[C{{'}_{j}}\] is column of second determinant i.e. B
(j,i<=3, because order of determinant is 3)
Similarly, if A and B is multiplied using point (iii), we have
\[A\times B=\left| \begin{matrix}
   {{R}_{1}}{{C}_{1}} & {{R}_{1}}{{C}_{2}} & {{R}_{1}}{{C}_{3}} \\
   {{R}_{2}}{{C}_{1}} & {{R}_{2}}{{C}_{2}} & {{R}_{2}}{{C}_{3}} \\
   {{R}_{3}}{{C}_{1}} & {{R}_{3}}{{C}_{2}} & {{R}_{3}}{{R}_{3}} \\
\end{matrix} \right|\]
Where, \[{{R}_{i}}\] is row of first determinant i.e. A
\[{{C}_{j}}\] is column of second determinant i.e. B
(j,i<=3, because order of determinant is 3)
And lastly, if point (iv) is used, we get
\[A\times B=\left| \begin{matrix}
   {{C}_{1}}{{R}_{1}} & {{C}_{2}}{{R}_{1}} & {{C}_{3}}{{R}_{1}} \\
   {{C}_{1}}{{R}_{2}} & {{C}_{2}}{{R}_{2}} & {{C}_{3}}{{R}_{2}} \\
   {{C}_{1}}{{R}_{3}} & {{C}_{2}}{{R}_{3}} & {{C}_{3}}{{R}_{3}} \\
\end{matrix} \right|\]
Where, \[{{C}_{i}}\] is column of first determinant i.e. A
\[{{R}_{j}}\] is row of second determinant i.e. B
(j,i<=3, because order of determinant is 3)
And, now we get a final result after multiplying A and B using row by row multiplication rule (i).
\[\text{Result}=\left| \begin{matrix}
   ap+bq+cr & as+bt+cu & av+bw+cx \\
   dp+eq+fr & ds+et+fu & dv+ew+fx \\
   gp+hq+ir & gs+ht+iu & gv+wh+ix \\
\end{matrix} \right|\]

Note: Therefore, we learnt 4 different rules of multiplication in determinants. Using the same concept as above for each rule, you will get the resultant determinant. If each element of a row or a column of a matrix is expressed as a sum of two or more terms, then the determinant can be calculated as the sum of two more determinants.