MINVERSE in Google Sheets: Invert Matrices and Check Linear Solutions
Use Sheets MINVERSE with a complete inverse, both identity products, a two-equation solve and explicit singularity and numerical limits.
Verification status: See the evidence and test scope described in this guide. How formula examples are checked
Quick answer: MINVERSE returns a multiplicative matrix inverse. For rows 3,1 and 2,1, the independently expected inverse has rows 1,−1 and −2,3. It is not the cell-by-cell reciprocal of the original entries.
Availability and version history
- Introduced: first service launch date is not established by the official history reviewed; no numbered desktop introduction is assigned.
- Availability: documented in the rolling Google Sheets service. Examples target ordinary web-spreadsheet cells; mobile clients and imported file modes are not separately certified.
- Checked: October 14, 2026; official MINVERSE reference. Documentation and independently calculated examples, without native Sheets execution.
Syntax and input shape
=MINVERSE(square_matrix)
Google requires a square array or range. A multiplicative inverse exists mathematically only for a nonsingular square matrix. The short function page does not assign every failure condition a verified native Sheets error code.
Use numeric values and a clear output area large enough for the resulting matrix. The result has the same square dimensions as the input. These are ordinary Sheets web-array controls, not an imported Excel legacy array-entry procedure.
Example 1: construct and invert a two-by-two matrix
Enter A2:B3 as follows:
| A | B |
|---|---|
| 3 | 1 |
| 2 | 1 |
In a separate clear two-by-two area beginning D2:
=MINVERSE(A2:B3)
Expected D2:E3 values are:
| D | E |
|---|---|
| 1 | -1 |
| -2 | 3 |
The determinant is 3×1−1×2=1. The two-by-two inverse formula swaps the diagonal entries, negates the off-diagonal entries and divides every result by the determinant, producing these four values.
The array-literal control =MINVERSE({3,1;2,1}) represents the same matrix in the shown English locale. Use cell ranges if array-literal separators are unfamiliar or differ with locale settings.
Example 2: verify multiplication in both orders
With A2:B3 and the expected inverse in D2:E3:
=MMULT(A2:B3,D2:E3)
=MMULT(D2:E3,A2:B3)
Enter these in separate clear two-by-two output areas. Both independently expected results are the identity, with rows 1,0 and 0,1.
For example, the first product’s upper-left entry is 3×1+1×(−2)=1, while its upper-right entry is 3×(−1)+1×3=0. Checking all four entries is stronger than checking only one diagonal value.
Advanced example: solve two equations with an explicit right-hand side
The original matrix represents:
- 3x+y=7
- 2x+y=5
Put 7 and 5 in G2:G3. In a clear two-row output area:
=MMULT(MINVERSE(A2:B3),G2:G3)
Expected solution values are x=2 and y=1. Substitution confirms 3×2+1=7 and 2×2+1=5. The residuals, calculated as reconstructed left sides minus the original right sides, are both 0 for this exact control.
A returned pair of numbers should be checked against the original equations, especially after changing a coefficient or accidentally reversing the right-hand-side rows. Matrix inversion is one algebraic method, not a claim that explicit inversion is the best numerical solver for every large system.
Advanced check: scale the whole matrix
Double every original entry to create rows 6,2 and 4,2. The inverse should be half the original inverse, with rows 0.5,−0.5 and −1,1.5.
The determinant of this doubled matrix is 4, not 2. The inverse product still gives the identity. If only one row is scaled, this whole-matrix halving shortcut does not apply uniformly to every inverse entry.
These controls distinguish matrix scaling from elementwise reciprocal operations. The reciprocal array for the original entries would contain 1/3, 1, 1/2 and 1, which does not have the expected identity-product property.
Advanced example: identify a singular system before claiming a solution
Rows 3,1 and 6,2 are dependent. Their determinant is 0. With right-hand side 7,14, the two equations represent the same line and have infinitely many solutions; with 7,15, they are inconsistent and have none.
Neither system has a unique solution that can be obtained by an ordinary inverse. The direct Google MINVERSE page does not establish a particular displayed error code for this control, so this guide states the mathematical limitation rather than borrowing another product’s code.
A very small nonzero determinant also does not guarantee a stable numerical calculation. Use MDETERM for a determinant control and inspect residuals, scaling and conditioning before relying on sensitive results.
Array display and evidence limits
Keep the requested output cells clear, verify the returned dimensions and retain the original input matrix. The dedicated source documents a square-matrix inverse but does not establish a complete blank/text/obstructed-output error table, maximum dimensions or precision bound.
Google’s ARRAYFORMULA reference discusses automatic expansion of many array formulas; it does not make every imported desktop entry convention portable. MMULT supplies the documented product operation used in the checks above.
References and verification
Inputs are fictional. Approximate answers are independent mathematical expectations rounded to nine places, not captured native results. Complete type/error matrices, extreme-magnitude accuracy and cross-client behavior remain untested.