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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