The Determinant of a 3×3 Matrix

Home Algebra Matrices and Determinants Determinant of a Matrix

After showing how to create the determinate of a matrix with order 3×3 in Minors and Cofactors of a Square Matrix, the following presents how to find the determinant more efficiently.

Matrix A

3×3	Column 1	Column 2	Column 3
Row 1:	0+	2−	1+
Row 2:	3−	−1+	2−
Row 3:	4+	0−	1+

First determine the minors for any row or column of array elements …

M11 = −1(1) − 0(2) = −1

M12 = 3(1) − 4(2) = −5

M13 = (3)(0) − 4(−1) = 4

Matrix A Row 1 Minors

M11 = −1

M12 = −5

M13 = 4

Next determine the cofactors by applying the matrix cofactor sign pattern to the minors of a row or column …

Matrix A (Row 1 Cofactors)

C11 = (+) −1 = −1

C12 = (−) −5 = 5

C13 = (+) 4 = 4

Use the cofactors and corresponding array elements from the row or column to compute the matrix determinant …

det(A) = 0(−1) + 2(5) + 1(4) = 14

Or, we could have chosen any other row or column (we’ll show column 2):

Matrix A (Column 2 Minors)

M12 = −5

M22 = −4

M32 = −3

Matrix A (Column 2 Cofactors)

C12 = (−) −5 = 5

C22 = (+) −4 = −4

C32 = (−) −3 = 3

det(A) = 2(5) + −1(−4) + 0(3) = 14

The determinant of matrix A is 14.

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The use of a determinant is algorithmic rather than mathematical and is important to solve for variable quantities of linear equation systems by Cramer’s Rule.

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