Determinant Formula
The determinant formula is used to find the determinant of a given matrix quickly. The determinant of a matrix is a number that is used in the analysis of linear equations and their solution. It is a mathematical object which is defined only for square matrices. A square matrix is a matrix in which the number of rows is equal to the number of columns. Let us learn about the determinant formula along with a few solved examples.
What is Determinant Formula?
The determinant of a matrix is a number that is used in the analysis of linear equations and their solution.The determinant formula for 2 by 2 matrix that is \(\left [\begin{matrix}a & b\\c & d\end{matrix}\right]\) is given by:
\(D_{2 \times 2}\)_{ }= ad  bc
The determinant formula for 3 by 3 matrix that is \(\left [\begin{matrix}a & b & c\\d & e & f\\ g & h & i\end{matrix}\right]\) is given by:
\(D_{3 \times 3}\)_{ }= a(eifh)b(difg)+c(dheg)
Let's take a quick look at a couple of examples to understand the determinant formula, better.
Solved Examples Using Determinant Formula

Example 1:Find the determinant of the 2x2 matrix below:
\(\left [\begin{matrix}2 & 3\\4 & 8\end{matrix}\right]\)Solution:
To find: Determinant of the matrix.
Given:
a = 2; b = 3
c = 4; d = 8Using determinant formula,
\(D_{2 \times 2}\) = ad  bc
Put the values,
\(D_{2 \times 2}\) = 2(8)3(4)
=1612
= 4
Answer: Determinant of the matrix is 4.

Example 2: Find the determinant of the 3x3 matrix below:
\(\left [\begin{matrix}6 & 1 & 1\\4 & 2 & 5\\ 2 & 8 & 7\end{matrix}\right]\) .Solution:
To find: Determinant of the matrix
a = 6; b = 1; c = 1
d = 4; e = 2; f = 5
g = 2; h = 8; i = 7 (given)Using determinant formula,
\(D_{3 \times 3}\) =a(ei  fh)  b(di  fg) + c(dh  eg)
=6(−2×7 − 5×8) − 1(4×7 − 5×2) + 1(4×8 − (−2×2))
= 6(−54) − 1(18) + 1(36)
= 306
Answer: Determinant of the matrix is (306).