Determinant of 3×3 Matrix

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Before reading this chapter, you will need to have the basic conepts of:

Determinant of a 3×3 Matrix is defined as the sum of products of each element and their respective Cofactors, taking one row or one column at a time.

Let us consider a matrix A of order 3×3 as

$\text{A} = \left ( \displaylines{a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}} \right )_{3\text{x}3}$

Then determinant of matrix A is defined as

$\text{|A|} \ = \ a_{11} \times C_{11} \ + \ a_{12} \times C_{12} \ + \ a_{13} \times C_{13} $

Also,

$\text{|A|} \ = \ a_{11} \times C_{11} \ + \ a_{21} \times C_{21} \ + \ a_{31} \times C_{31} $

$\text{|A|} \ = \ a_{21} \times C_{21} \ + \ a_{22} \times C_{22} \ + \ a_{23} \times C_{23} $

Thus, determinant of a matrix is obtained by expanding the matrix along any row or column of the given matrix, taking only one at a time.

Solution:

Given,

$\text{B} = \left ( \displaylines{2 & 5 & 10 \\ 3 & 6 & 11 \\ 4 & 7 & 12} \right )_{3\text{x}3}$

STEP 1: Select a row or a column that you want to expand.

STEP 2: Find the minor of each element present in the selected row or column in STEP 1.

STEP 3: Find the cofactor of each element present in the selected row or column in STEP 1.

STEP 4: Calculate the sum of the product of each elements and their respective cofactors of the selected row or column.

STEP 5: The obtained value in STEP 4 is the required value of determinant of 3×3 matrix.

Applications of Determinant of 3×3 Matrix