# Trick to calculate determinant of a 3×3 matrix

Linear Algebra is one of the most important topics in Engineering Mathematics Gate syllabus. Finding determinant of a matrix is one of the most important problems in Linear Algebra. Finding Determinant of a matrix is required for finding inverse of a matrix, determining whether vectors are linearly independent or not etc.

Let us consider a matrix and its determinant be A, then A can be calculated as given below.

where,

Example :

```A = 1( 5*9 â€“ 6*8) â€“ 2(4*9 â€“ 6*7) + 3(4*8 â€“ 5*7)
A = 1(45 â€“ 48) â€“ 2(36 â€“ 42) + 3(32 â€“ 35)
A = 1*(-3) â€“ 2*(-6) + 3*(-3)
A = -3 + 12 â€“ 9
A = 0 ```

The above traditional method consumes a lot of time especially when you are solving some complex problem. There is another easier and faster method determinant of a matrix. The below shown is faster way of solving the determinant of a matrix.

Another Method for faster calculation :
In this method we represent the matrix in different method only for determinant calculation.

Consider a matrix and convert it into the following given below.

write the matrix as

Now perform the following operation,

Here,

So, from the above matrix, we can write,

```i = (3*4*8) = 96     x = (2*4*9) = 72
j = (1*5*9) = 45     y = (3*5*7) = 105
k = (2*6*7) = 84     z = (1*6*8) = 48

A = (i + j + k) â€“ (x + y + z) = (96 + 45 + 84) â€“ (72 + 105 + 48)
= (225 â€“ 225)
= 0 ```

So, by following the above we can calculate the determinant of a matrix easily. It requires practice to change our method of calculation from traditional method to easy method but it is worth practicing,

Note –
This method works only for (3, 3) matrix.