Tuesday, February 5

Learning to find the determinant value of a 3x3 matrix



Consider a matrix A. If A is a square matrix then there is some value that can be assigned to A. This is called the determinant of matrix A. It is denoted by the symbol |A| or det(A). The entries of the matrices are used and an arithmetic expression is formed out of them to come to a value of the matrix. Such square matrices are most commonly used in solving system of linear equations. If the number of variables in the system is 2, then the square matrix would be a 2x2 matrix. Similarly if the number of variables in the system is n, then the square matrix would be an nXn matrix.
Determinant of a 3x3 matrix:
The determinant of a 3x3 matrix is also called a third order determinant. If we have to solve a system of three simultaneous linear equations in three variables, we shall have to deal with third order determinants or 3x3 determinant. For real numbers a1, a2, a3, b1, b2, b3, c1, c2, c3, the symbol
|a1 a2 a3|
|b1 b2 b3|
|c1 c2 c3|

Represents the determinant of 3x3 matrix.

The numbers a1, a2, a3, b1, b2, b3, c1, c2, c3 are the entries or elements of the det.  The first, the second and the third rows of the det are respectively: a1, a2, a3; b1, b2, b3 and c1, c2, c3. The first, the second and the third columns are respectively;
a1 a2 a3
b1; b2; b3;
c1 c2 c3

The value of this determinant of 3x3 matrix is defined as:

|a1 a2 a3|
|b1 b2 b3|
|c1 c2 c3|

= a1*|b2 b3| - a2 * |b1 b3| + a3 * |b1 b2|
          |c2 c3|            |c1 c3|             |c1 c2|

= a1 * (b2c3 – b3c2) – a2 * (b1c3 – b3c2) + a3 * (b1c2 – b2c1)

The above formula can be used to find the value of any 3x3 determinant. This is not the only method used to evaluate a 3x3 determinant. We can also use the Sarrus’ expansion method. It is as follows:
Suppose D
= |a b c|
   |d e f|
   |g h i|
We write five columns like this:

a b c a b
d e f d e
g h I g h
Then we multiply the terms long the red and blue lines as shown in the picture below:

The red diagonals are aei, bfg and cdh. The blue diagonals are gec, hfa and idb.
Then D = aei + bfg + cdh – gec – hfa - idb

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