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Question: How do you find the determinant of \[\left( \begin{matrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ ...

How do you find the determinant of (123 456 789 )\left( \begin{matrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9 \\\ \end{matrix} \right)?

Explanation

Solution

Assume the given matrix as ‘A’. Then check if it contains the same number of rows and columns or not. If it has same numbers of rows and columns then the determinant can be calculated using the formula ∣A∣=a(ei−hf)−b(di−gf)+c(dh−ge)\left| A \right|=a\left( ei-hf \right)-b\left( di-gf \right)+c\left( dh-ge \right), where the determinant value of ‘A’ is denoted as ∣A∣\left| A \right|.

Complete step by step solution:
Determinant: The determinant of a matrix is a special number that can be calculated from a square matrix.
Calculating the determinant: If a matrix has the same number of rows and columns i.e. square matrix then it’s determinant can be calculated. For a 3×33\times 3 square matrix A=(abc def ghi )A=\left( \begin{matrix} a & b & c \\\ d & e & f \\\ g & h & i \\\ \end{matrix} \right), the determinant is denoted as ∣A∣\left| A \right| which can be calculated as ∣A∣=a(ei−hf)−b(di−gf)+c(dh−ge)\left| A \right|=a\left( ei-hf \right)-b\left( di-gf \right)+c\left( dh-ge \right)
Assuming the given matrix as A=(123 456 789 )A=\left( \begin{matrix} 1 & 2 & 3 \\\ 4 & 5 & 6 \\\ 7 & 8 & 9 \\\ \end{matrix} \right)
Since it has same numbers of rows and columns i.e. ‘3’
So, by comparison we get a=1, b=2, c=3, d=4, e=5, f=6, g=7, h=8 and i=9
Hence the determinant of ‘A’ can be calculated as
∣A∣=1(5×9−8×6)−2(4×9−7×6)+3(4×8−7×5) =1(45−48)−2(36−42)+3(32−35) =1×(−3)−2×(−6)+3×(−3) =−3−(−12)−9 =−3+12−9 =0 \begin{aligned} & \left| A \right|=1\left( 5\times 9-8\times 6 \right)-2\left( 4\times 9-7\times 6 \right)+3\left( 4\times 8-7\times 5 \right) \\\ & =1\left( 45-48 \right)-2\left( 36-42 \right)+3\left( 32-35 \right) \\\ & =1\times \left( -3 \right)-2\times \left( -6 \right)+3\times \left( -3 \right) \\\ & =-3-\left( -12 \right)-9 \\\ & =-3+12-9 \\\ & =0 \\\ \end{aligned}
This is the required solution of the given question.

Note:
The determinant can be calculated by solving any row or column. While solving any row or column the sign convention should be remembered as (\+\-\+ \-\+\- \+\-\+ )\left( \begin{matrix} \+ & \- & \+ \\\ \- & \+ & \- \\\ \+ & \- & \+ \\\ \end{matrix} \right). Other applications of determinant are to find the inverse of a matrix, solving system of linear equations etc.