**MDETERM Function in Microsoft Excel**

### Part 1: Introduce

**Definition:**

The MDETERM function in Microsoft Excel returns the matrix determinant of an array.

**Purpose:**

Matrix determinants are generally used for solving mathematical equation systems involving several variables.

**Syntax & Arguments:**

`MDETERM(array)`

**Explain the Arguments in the function:**

**Array:**Required. A numeric array with an equal number of rows and columns. This array can be given as a cell range (e.g., A1:C3), as an array constant (e.g., {1,2,3;4,5,6;7,8,9}), or as a name to either of these.

**Return value:**

The matrix determinant is a number derived from the values in the array.

**Remarks:**

- MDETERM returns the
`#VALUE!`

Error when:- Any cells in the array are empty or contain text.
- The array does not have an equal number of rows and columns.

- MDETERM is calculated with an accuracy of approximately 16 digits, which may lead to a small numeric error when the calculation is incomplete.

### Part 2: Examples

🔹 **Example 1:****Purpose of example:** Determinant of a 3×3 matrix.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 3 | 8 | 5 |

2 | 3 | 6 | 1 |

3 | 1 | 1 | 0 |

=MDETERM(A1:C3) | Result | 88 |

**Explanation:**

This example calculates the determinant of a 3×3 matrix using the MDETERM function.

🔹 **Example 2:****Purpose of example:** Determinant of a matrix using array constants.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 3 | 6 | 1 |

2 | 1 | 1 | 0 |

3 | 3 | 10 | 2 |

=MDETERM({3,6,1;1,1,0;3,10,2}) | Result | 1 |

**Explanation:**

This example demonstrates using array constants directly in the MDETERM function to calculate the determinant.

🔹 **Example 3:****Purpose of example:** Determinant of a 2×2 matrix using array constants.**Data sheet and formulas:**

A | B | |
---|---|---|

1 | 3 | 6 |

2 | 1 | 1 |

=MDETERM({3,6;1,1}) | Result |

**Explanation:**

This example calculates the determinant of a 2×2 matrix using array constants.

🔹 **Example 4:****Purpose of example:** Error due to unequal rows and columns.**Data sheet and formulas:**

A | B | C | D | |
---|---|---|---|---|

1 | 1 | 3 | 8 | 5 |

2 | 1 | 3 | 6 | 1 |

=MDETERM({1,3,8,5;1,3,6,1}) | Result | #VALUE! |

**Explanation:**

This example returns an error because the array does not have an equal number of rows and columns.

🔹 **Example 5:****Purpose of example:** Determinant of a 4×4 matrix.**Data sheet and formulas:**

A | B | C | D | |
---|---|---|---|---|

1 | 1 | 3 | 8 | 5 |

2 | 1 | 3 | 6 | 1 |

3 | 7 | 3 | 10 | 2 |

=MDETERM(A1:D3) | Result | 88 |

**Explanation:**

This example calculates the determinant of a 4×4 matrix using the MDETERM function.

🔹 **Example 6: Using MDETERM with IF****Purpose of example:** Determine if the matrix determinant is positive or negative.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 2 | 5 | 3 |

2 | 4 | 1 | 6 |

3 | 7 | 3 | 2 |

=IF(MDETERM(A1:C3)>0, “Positive”, “Negative”) | Result | Positive |

**Explanation:**

This example checks if the determinant of the matrix is positive or negative using the IF function. If the determinant is positive, it returns “Positive”; otherwise, it returns “Negative”.

🔹 **Example 7: Using MDETERM with SUM****Purpose of example:** Sum the matrix determinant with another value.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 3 | 2 | 4 |

2 | 5 | 6 | 1 |

3 | 8 | 7 | 3 |

=SUM(MDETERM(A1:C3), 10) | Result | 52 |

**Explanation:**

This example calculates the determinant of the matrix and then adds 10 to the result using the SUM function.

🔹 **Example 8: Using MDETERM with VLOOKUP****Purpose of example:** Find the determinant value from a lookup table.**Data sheet and formulas:**

A | B | C | D | E | F | |
---|---|---|---|---|---|---|

1 | 1 | 2 | 3 | 45 | “Matrix A” | |

2 | 4 | 5 | 6 | 60 | “Matrix B” | |

3 | 7 | 8 | 9 | 0 | “Matrix C” | |

=VLOOKUP(MDETERM(A1:C3), E1:F3, 2, FALSE) | Result | Matrix A |

**Explanation:**

This example uses the VLOOKUP function to find the name of the matrix based on its determinant value from a lookup table.

🔹 **Example 9: Using MDETERM with AVERAGE****Purpose of example:** Average the matrix determinant with another set of numbers.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 2 | 4 | 6 |

2 | 8 | 5 | 3 |

3 | 7 | 1 | 9 |

=AVERAGE(MDETERM(A1:C3), 5, 10, 15) | Result | 8.75 |

**Explanation:**

This example calculates the average of the matrix determinant and three other numbers (5, 10, 15) using the AVERAGE function.

🔹 **Example 10: Using MDETERM with MAX****Purpose of example:** Find the maximum value between the matrix determinant and another set of numbers.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 3 | 6 | 9 |

2 | 2 | 5 | 8 |

3 | 1 | 4 | 7 |

=MAX(MDETERM(A1:C3), 10, 20, 30) | Result | 30 |

**Explanation:**

Using the MAX function, this example determines the maximum value between the matrix determinant and three other numbers (10, 20, 30).

🔹 **Example 11: Using MDETERM with MIN****Purpose of example:** Find the minimum value between the matrix determinant and another set of numbers.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 1 | 3 | 5 |

2 | 7 | 9 | 2 |

3 | 4 | 6 | 8 |

=MIN(MDETERM(A1:C3), 5, 10, 15) | Result | 5 |

**Explanation:**

Using the MIN function, this example determines the minimum value between the matrix determinant and three other numbers (5, 10, 15).

🔹 **Example 12: Using MDETERM with ROUND****Purpose of example:** Round the matrix determinant to the nearest whole number.**Data sheet and formulas:**

A | B | C | |
---|---|---|---|

1 | 2 | 5 | 8 |

2 | 3 | 6 | 9 |

3 | 4 | 7 | 1 |

=ROUND(MDETERM(A1:C3), 0) | Result | 27 |

**Explanation:**

This example rounds the determinant of the matrix to the nearest whole number using the ROUND function.

### Part 3: Tips and tricks

- Ensure that the array you provide has an equal number of rows and columns to avoid the
`#VALUE!`

error. - You can use cell ranges, array constants, or names to provide the array to the MDETERM function.
- Remember that the matrix determinant is a single number derived from the values in the array, and it’s used for solving systems of mathematical equations.