Question
Download Solution PDFWith the numbers 2, 4, 6, 8, all the possible determinants with these four different elements are constructed. What is the sum of the values of all such determinants?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
Number of ways of arranging n object = n!
Calculation:
Given numbers are 2, 4, 6 & 8.
We can form determinant of order 2.
We know that the number of ways of arranging n object = n!
Number of determinants = 4! = 4 × 3 × 2 × 1 = 24
Let's form some determinants from the given numbers.
\(\begin{vmatrix} 2 & \rm 6 \\ 8 & 4 \end{vmatrix}\) = 2 × 4 - 6 × 8 = 8 - 48 = - 40
\(\begin{vmatrix} 6 & \rm 2 \\ 4 & 8 \end{vmatrix}\) = 48 - 8 = 40
\(\begin{vmatrix} 2 & \rm 8 \\ 6 & 4 \end{vmatrix}\) = 8 - 48 = - 40
\(\begin{vmatrix} 6 & \rm 4 \\ 2 & 8 \end{vmatrix}\) = 48 - 8 = 40
\(\begin{vmatrix} 4 & \rm 8 \\ 6 & 2 \end{vmatrix}\) = 8 - 48 = - 40
\(\begin{vmatrix} 8 & \rm 4 \\ 2 & 6 \end{vmatrix}\) = 48 - 8 = 40
\(\begin{vmatrix} 4 & \rm 6 \\ 8 & 2 \end{vmatrix}\) = 8 - 48 = - 40
\(\begin{vmatrix} 8 & \rm 2 \\ 4 & 6 \end{vmatrix}\) = 48 - 8 = - 40
Hence, we can see that we are getting the pattern where the other value will neutralize each determinant value.
Accordingly, the sum of the values of all determinants = 0
∴ The required sum is zero.
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