Zoho Sheet MDETERM: Matrix Determinants, Signs and Scaling
Calculate a square matrix determinant in Zoho Sheet with complete two-by-two and three-by-three examples, row-change checks and scale-aware limits.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
MDETERM returns one number: the determinant of a square numeric matrix. For a two-by-two matrix, it is the first diagonal product minus the second. It helps explain whether a matrix has an inverse, but its magnitude also depends on the scale of the entries.
Availability and version history
- Introduced: The first introduction release and date are not established by the checked references.
- Availability: Listed in the current Zoho Sheet function help; no final supported-version endpoint is established.
- Scope: Current web-service documentation. Historical builds, mobile clients, offline use and imported workbooks remain separately unverified.
- Checked: October 14, 2026, against the official MDETERM reference. All worked outputs were independently calculated; no native Zoho Sheet test was performed.
Syntax and the square-matrix requirement
MDETERM(array)
The MDETERM reference accepts a numeric array or cell range representing a square matrix. A two-row input must therefore also have two columns; a three-row input needs three columns. The single argument has no documented default.
Select only the numeric block. Row labels, column headers and a neighboring total are not part of the matrix. The result is scalar, so this function does not need a multi-cell output area.
Calculate a two-by-two determinant
Create a sheet named TwoByTwo, reserving A1:F7. Put First coefficient and Second coefficient in A1:B1, then enter:
| Row | A | B |
|---|---|---|
| 2 | 3 | 1 |
| 3 | 2 | 4 |
Enter =MDETERM(A2:B3) in D2. The independently calculated result is 10. In D3 enter =A2*B3-B2*A3; it also gives 10, because 3×4−1×2=12−2. Put =D2-D3 in D4 for an expected difference of 0.
The explicit two-by-two calculation checks entry and selection mistakes. It cannot detect every mistake: transposing a square matrix preserves its determinant. For this example, rows 3,2 and 1,4 still give 10 even though their positions differ.
Zoho’s MINVERSE reference requires a nonzero determinant for an inverse. The value 10 therefore satisfies that mathematical prerequisite; it is not a precision guarantee for every numerical computation.
Change one structural feature at a time
Keep TwoByTwo unchanged. Make a separate copy for each row below, replacing all four A2:B3 entries with the stated values. D2:D4 retain their original formulas.
| Sheet | Row 2: A, B | Row 3: A, B | Expected D2 |
|---|---|---|---|
| SwapRows | 2, 4 | 3, 1 | -10 |
| RepeatRow | 3, 1 | 3, 1 | 0 |
| DoubleFirstRow | 6, 2 | 2, 4 | 20 |
| DoubleEveryEntry | 6, 2 | 4, 8 | 40 |
| Transposed | 3, 2 | 1, 4 | 10 |
Swapping the rows reverses the sign. Repeating a row gives a singular matrix: the two rows supply dependent information. Doubling one row doubles the determinant, while doubling every entry in this two-by-two example multiplies it by four.
These controls separate a sign change, a loss of invertibility and a change of units. A negative determinant is not an error; it does not mean the matrix lacks an inverse.
Advanced: verify a three-by-three expansion
Create ThreeByThree, using A1:F8. Put coefficient headings in A1:C1 and enter the following numeric block:
| Row | A | B | C |
|---|---|---|---|
| 2 | 2 | 1 | 0 |
| 3 | 0 | 3 | 1 |
| 4 | 1 | 0 | 2 |
Put =MDETERM(A2:C4) in E2. Expect 13. Independently reconstruct the first-row expansion in E3:
=A2*(B3*C4-C3*B4)-B2*(A3*C4-C3*A4)+C2*(A3*B4-B3*A4)
The three signed contributions are 12,1,0, summing to 13. In E4 put =E2-E3 for an expected zero. The middle term has a subtraction sign before a negative minor, so dropping that sign would produce the wrong answer.
For another copy named ScaledThree, replace each of the nine input entries with twice its original value. The first row becomes 4,2,0; the second 0,6,2; the third 2,0,4. E2 and E3 now have the expected value 104, eight times 13. A three-by-three determinant scales by the cube when every entry is multiplied by the same factor.
Small determinants and practical limits
On a new SmallScale sheet, enter row 2 0.001,0 and row 3 0,0.001 in A:B. Put =MDETERM(A2:B3) in D2 for the mathematical result 0.000001. This diagonal matrix is invertible despite the small determinant. Its inverse has diagonal entries 1000; the determinant is small because of the scale chosen.
Do not use a universal rule such as “determinant below 0.01 means unusable.” Determinant magnitude alone does not establish numerical sensitivity. For a calculated inverse, multiplying back with MMULT provides a useful residual check, without proving that rounded input data are accurate.
The short official entry gives generic error descriptions but does not identify a special error token for every nonsquare, empty or mixed-type matrix. A singular square matrix has determinant zero mathematically; its failure to have an inverse is a separate issue. Extreme size, numerical tolerance and underflow behavior remain unverified.
The examples contain ordinary numeric cells and use semicolons between multiple arguments, matching the English function references. Check the formula separators in your workbook. Source blanks, numeric text, Boolean values, existing error cells and imported array formulas are outside the tested mathematical fixtures.