Zoho Sheet MINVERSE: Invert Matrices and Check Linear Solutions
Use Zoho Sheet MINVERSE with complete matrix inputs, documented array placement, multiplication checks and a two-equation solution with residuals.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
MINVERSE returns the inverse of a square matrix. Multiplying the original matrix by its inverse gives an identity matrix in exact arithmetic. This makes it useful for small instructional systems of simultaneous equations, provided the matrix is invertible.
The inverse is not obtained by replacing every entry with its reciprocal. The example below has negative inverse entries even though all four original entries are positive.
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 MINVERSE reference. All worked outputs were independently calculated; no native Zoho Sheet test was performed.
Syntax, output shape and prerequisites
MINVERSE(array)
The MINVERSE reference specifies a square numeric array or range and states that its determinant must be nonzero. The inverse has the same dimensions as the original. There is one argument and no documented default.
Zoho’s dynamic-array instructions describe entering an array formula in one cell with ordinary Enter. Leave its output rectangle empty; an obstruction produces #SPILL!. Enter only the anchor formula, rather than filling copies across the output. These exercises follow that documented web workflow.
Invert a two-by-two matrix and multiply back
Create InverseCheck, with working space A1:M12. Put coefficient headings in A1:B1, and enter rows 3,1 and 2,4 in A2:B3. Keep F2:G3, I2:J3 and L2:M3 empty. Labels can go in F1, I1 and L1, above those output areas.
- In D2 enter
=MDETERM(A2:B3). Its independent result is 10. - In F2 enter
=MINVERSE(A2:B3)and press Enter. The expected F2:G3 grid is shown below. - In I2 enter
=MMULT(A2:B3;F2:G3). - In L2 enter
=MMULT(F2:G3;A2:B3).
| Row | F: inverse column 1 | G: inverse column 2 | I | J | L | M |
|---|---|---|---|---|---|---|
| 2 | 0.4 | -0.1 | 1 | 0 | 1 | 0 |
| 3 | -0.2 | 0.3 | 0 | 1 | 0 | 1 |
The hand formula is 1/10 times the matrix with rows 4,-1 and -2,3. Checking both multiplication orders makes the role of the inverse explicit. MUNIT generates an identity matrix for other comparison exercises.
These are exact mathematical outputs. A native numerical calculation may display small residuals instead of exact zeros. Inspect sufficient decimal places and consider input precision; do not silently round away a large discrepancy.
Advanced: solve two simultaneous equations
On the same sheet, keep the original matrix and inverse intact. Enter 11 in D6 and 24 in D7. These are the right-hand sides of:
- 3x+y=11
- 2x+4y=24
Leave F6:F7 and H6:H7 empty. In F6 enter =MMULT(F2:G3;D6:D7). The expected solution is x=2 in F6 and y=5 in F7.
In H6 enter =MMULT(A2:B3;F6:F7) to reconstruct 11,24. Put =H6-D6 in J6 and fill down to J7. Both residuals have an expected value of 0. The two explicit substitutions, 3×2+5=11 and 2×2+4×5=24, provide an independent check.
For a separate scenario, copy the sheet as ChangedTarget and change only D7 to 25. The matrix and its inverse do not change. The new solution is 1.9,5.3, and H6:H7 reconstructs 11,25. J6:J7 again has expected zeros.
The solution changes by -0.1,+0.3. That is the second column of the inverse multiplied by the one-unit increase in the second right-hand side. A change in the target vector is different from changing the coefficients.
A three-by-three check with fractional entries
Create ThreeInverse, reserving A1:L7. Enter rows 2,1,0, 0,3,1, 1,0,2 in A2:C4. Put =MDETERM(A2:C4) in E2 for 13. Leave F2:H4 and J2:L4 empty.
In F2 enter =MINVERSE(A2:C4). The exact expected inverse is:
| Output row | F | G | H |
|---|---|---|---|
| 2 | 6/13 | -2/13 | 1/13 |
| 3 | 1/13 | 4/13 | -2/13 |
| 4 | -3/13 | 1/13 | 6/13 |
The fractions describe the expected numbers; do not type them over the output cells. In J2 enter =MMULT(A2:C4;F2:H4). Expect the three-by-three identity, with ones at J2,K3,L4 and zeros elsewhere.
If you need decimal displays, 6/13 is approximately 0.4615384615, 1/13 approximately 0.0769230769, and -2/13 approximately -0.1538461538. Formatting the results does not make the input or inversion exact.
Singular inputs and troubleshooting
The matrix with rows 3,1 and 3,1 has determinant zero and no inverse. Do not promise a particular native error token from the short MINVERSE entry: it states the prerequisite but does not give a complete singularity/error table.
If an array result is obstructed, inspect the planned output rectangle before clearing anything. Move unrelated data out of the way or choose a new empty anchor. An inherited @ operator can suppress an array result in Zoho’s documented transition behavior; compare the actual formula with these newly entered exercises.
For larger or sensitive systems, a hand-built inverse is not a substitute for an appropriately selected numerical solver. The examples do not establish an algorithm, condition threshold, maximum matrix size or cross-product precision guarantee.
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.
MDETERM checks the determinant. MMULT performs matrix multiplication, while reciprocal arithmetic on individual cells answers a different question.