Determinants of 2 x 2 matrices are easy to calculate.
A square matrix of order higher than 2 is a 3 x 3 matrix or higher. For example, 4 x 4 matrix, 5 x 5 matrix, 6 x 6 matrix, etc…
To solve square matrices of order higher than 2 use the Definition of the determinant of a square matrix.
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Definition of the determinant of a square matrix
If A is a square matrix of order n≥2, then the determinant of A is the sum of the entries in the first row of A multiplied by their respective cofactors. This is referred to as first row expansion or expansion by the cofactors in the first row.
det(A) = |A|=
The subscript numbers are the row number and column number.
Even though the definition says to expand using the first row you can expand in any direction using any row or column and still get the same answer.
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Example of an 3 x 3 matrix:
Find the determinant of
Expand by the cofactors of the first row:
Step 1. Cancel row 1 and column 1.
Step 2. Find the minor of row 1, column 1. M11 = (4)(1) – (2)(5) = 4 – 10 = 6
Step 3. Find the cofactor by multiply the minor by the appropriate sign of pivot number position.
Step 4. Repeat steps 1-3 for Row 1 Column 2 and Row 1 Column 3
|A|= 18 + 3 + 0 = 21