Determinants develop from number patterns that emerge when solving systems of linear equations. Every square matrix can be associated with a negative, positive or zero real number determinant. 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.

The reason for introducing minors and cofactors of a square matrix is to develop a constructive foundation necessary to create a matrix determinant.

The minor, Mij, of the entry aij is the determinant of the matrix. It is obtained by deleting the j th row and the i th column of A.

The cofactor Cij of entry aij is:

Cij = (−1) i + j Mij

Cofactors have a sign pattern:

Matrix Cofactor Sign Pattern

4×4 |
C1 |
C2 |
C3 |
C4 |

R1: |
+ |
− |
+ |
− |

R2: |
− |
+ |
− |
+ |

R3: |
+ |
− |
+ |
− |

R4: |
− |
+ |
− |
+ |

The sign pattern begins as a “+” top left and then alternates as “−/+” thereafter for each element of the row. The next row beneath the first begins with a “−”, and then alternates “+/−”. The alternating pattern continues for the entire matrix.

Finding the Minors and Cofactors of a Matrix

To find the minors and cofactors (we’ll visually present the first two minors so you can easily understand the process):

The first minor M11 …

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+ |

M11

−1 |
2 |

0 |
1 |

=

−1(1) − 0(2) = −1

(Diagonal multiplication and then product subtraction)

The second minor M12 …

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+ |

M12

3 |
2 |

4 |
1 |

=

3(1) − 4(2) = −5

The remaining minors are determined using the same rule-based process …

M11 = −1

M21 = 2

M31 = 5

M12 = −5

M22 = −4

M32 = −3

M13 = 4

M23 = −8

M33 = −6

Then, to find the cofactors, apply the matrix cofactor sign pattern to the minors …

C11 = −1

C21 = −2

C31 = 5

C12 = 5

C22 = −4

C32 = 3

C13 = 4

C23 = 8

C33 = −6

Expanding by Cofactors to Find the Matrix Determinant

For any matrix of order 2×2 or greater the determinant is the sum of the elements in any row or column multiplied by their respective cofactors:

det(A) = a11C11 + a12C12 + . . . + a1nC1n

By this definition, to find the determinant of A:

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+ |

We can choose row 1 with array elements 0, 2, 1, multiply each element by its cofactor and then sum the products of the multiplication (a row or column with the most zeros):

det(A) = 0(−1) + 2(5) + 1(4) = 14

Or, we could choose column 2 with array elements 2, 1, 0:

det(A) = 2(5) + −1(−4) + 0(3) = 14

The determinant of matrix A is 14.

Now you see the role minors and cofactors play in determining a matrix determinant and how to produce the determinant. However, because it is important to understand how to find a determinant the following information provides matrices of order 1×1 and 2×2, each with a detailed description of the process for finding their determinant.

The determinant of a matrix with order 1×1 is the entry of the matrix, the array element value: Matrix A = [−5] then det(A) = −5

The determinant of a 2×2 matrix is produced by subtracting after diagonal multiplication of array elements. It is the matrix formula and only works for a 2×2 matrix.

Matrix A

a1 |
b1 |

a2 |
b2 |

det(A) = a1b2 − a2b1

Matrix B

0 |
½ |

2 |
4 |

det(B) = 0 − 1 = −1

How to find the determinant of a 3×3 or 4×4 matrix can be found by clicking on tabs near the top of this page.

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