Master Matrix Multiplication with Excel's MMULT Formula
=MMULT(array1, array2)The MMULT function is a powerful mathematical tool in Excel that performs matrix multiplication, a fundamental operation in linear algebra and advanced data analysis. Matrix multiplication is essential for professionals working with complex datasets, financial modeling, engineering calculations, and statistical analysis. Unlike simple cell-by-cell multiplication, MMULT combines entire arrays of numbers following specific mathematical rules, enabling sophisticated calculations that would be extremely tedious to perform manually. This advanced function is particularly valuable for data scientists, financial analysts, and engineers who need to work with multidimensional data structures. MMULT requires understanding of matrix dimensions and mathematical principles, making it an intermediate-to-advanced Excel skill. When mastered, MMULT dramatically streamlines complex calculations, reduces formula complexity, and minimizes errors in array-based computations. Whether you're performing portfolio analysis, solving systems of equations, or transforming coordinate systems, MMULT provides an elegant and efficient solution for matrix operations.
Syntax & Parameters
The MMULT function follows the syntax =MMULT(array1, array2), where both parameters are required. The first parameter, array1, represents the first matrix you want to multiply. This matrix must have dimensions of m×n (m rows and n columns). The second parameter, array2, represents the second matrix, which must have dimensions of n×p (n rows and p columns). Critically, the number of columns in array1 must equal the number of rows in array2—this is a fundamental rule of matrix multiplication that Excel strictly enforces. The resulting matrix from MMULT will have dimensions of m×p, combining the row count from array1 with the column count from array2. Each element in the result matrix is calculated by multiplying corresponding elements from array1's rows with array1's columns from array2, then summing these products. When entering MMULT, you must use Ctrl+Shift+Enter to confirm it as an array formula in Excel 2019 and earlier versions. In Excel 365, the formula automatically spills results into adjacent cells. Both array1 and array2 should contain only numeric values; text or blank cells will cause errors.
array1array2Practical Examples
Portfolio Returns Calculation
=MMULT({0.4; 0.35; 0.25}, {0.08; 0.12; 0.06})The weights array contains portfolio allocations (40%, 35%, 25%) for three assets. The returns array contains expected returns (8%, 12%, 6%). MMULT multiplies these matrices to produce the overall portfolio expected return of 0.0845 or 8.45%.
Sales Forecast by Region and Product
=MMULT(B2:C4, E2:H3)The first matrix (B2:C4) contains sales units for 3 regions across 2 product categories. The second matrix (E2:H3) contains pricing for 2 categories across 4 product types. The result is a 3×4 matrix showing revenue for each region-product combination.
Transformation Matrix for Coordinate Systems
=MMULT(A1:C3, D1:F100)The rotation matrix (A1:C3) is a 3×3 transformation matrix. The coordinate data (D1:F100) represents 100 points with x, y, z coordinates. MMULT applies the rotation transformation to all points simultaneously, producing transformed coordinates.
Key Takeaways
- MMULT performs true matrix multiplication where column count of first array must equal row count of second array
- Result dimensions are determined by rows from array1 and columns from array2, producing an m×p matrix from m×n and n×p inputs
- Ctrl+Shift+Enter confirmation is required in Excel 2019 and earlier; Excel 365 handles array formulas automatically
- MMULT is superior to SUMPRODUCT for full matrix operations but SUMPRODUCT is simpler for single-value calculations
- Combine MMULT with TRANSPOSE, MINVERSE, or MUNIT for advanced linear algebra operations like solving equations or coordinate transformations
Pro Tips
Always verify matrix dimensions before writing MMULT formulas using ROWS() and COLUMNS() functions in helper cells.
Impact : Prevents dimension mismatch errors and saves debugging time. A quick dimension check can be done with =ROWS(array1)=COLUMNS(array2) to validate compatibility.
Use named ranges for matrix arrays to make MMULT formulas more readable and maintainable, especially in complex workbooks.
Impact : Improves formula clarity, reduces errors when modifying ranges, and makes your spreadsheet self-documenting. Instead of =MMULT(A1:C3, D1:F5), use =MMULT(WeightMatrix, ReturnsMatrix).
In Excel 365, use the SEQUENCE function to generate test matrices quickly: =MMULT(SEQUENCE(3,2), SEQUENCE(2,4)) creates sample 3×4 result without manual data entry.
Impact : Accelerates testing and prototyping of MMULT formulas. SEQUENCE generates sequential numbers, useful for validating formula logic before applying to real data.
Document your matrix structure (dimensions, row/column labels, units) in adjacent cells or comments to prevent calculation errors and aid collaboration.
Impact : Reduces mistakes when others modify spreadsheets, ensures correct interpretation of results, and serves as reference documentation for complex models.
Useful Combinations
MMULT with TRANSPOSE for Complex Transformations
=MMULT(TRANSPOSE(A1:C3), D1:D3)Combining MMULT with TRANSPOSE allows you to flip matrix dimensions before multiplication. This is essential when your data is arranged in rows but matrix operations require columns. The TRANSPOSE function converts a 3×3 matrix to 3×3 (in this case, same dimensions) but with rows and columns swapped, enabling different multiplication patterns.
MMULT with MINVERSE for Solving Linear Systems
=MMULT(MINVERSE(A1:C3), D1:D3)MMULT combined with MINVERSE solves systems of linear equations. MINVERSE calculates the inverse of a coefficient matrix, then MMULT multiplies it by the constants vector. This approach efficiently solves Ax=b equations, where A is the coefficient matrix, x is the solution vector, and b is the constants vector.
MMULT with MUNIT for Identity Matrix Operations
=MMULT(A1:C3, MUNIT(3))Multiplying any matrix by an identity matrix (created with MUNIT) returns the original matrix unchanged. This combination is useful for validation testing, creating baseline calculations, or understanding matrix properties. MUNIT(3) creates a 3×3 identity matrix, which is the matrix equivalent of multiplying by 1.
Common Errors
Cause: Array1 or array2 contains non-numeric values, text, or empty cells that cannot be processed in matrix multiplication.
Solution: Verify all cells in both matrices contain only numeric values. Use ISNUMBER() to check cells before MMULT. Remove or replace any text, spaces, or error values with appropriate numbers.
Cause: The array references are incorrect or point to deleted cells. Commonly occurs when copying formulas with relative references that shift beyond data boundaries.
Solution: Use absolute references ($A$1:$C$3) instead of relative references. Verify array ranges haven't been deleted or moved. Rebuild the formula with correct cell references.
Cause: The number of columns in array1 doesn't equal the number of rows in array2, violating matrix multiplication rules. Excel may not always display an error but produces incorrect calculations.
Solution: Count columns in array1 and rows in array2—these must be equal. For array1 with dimensions m×n and array2 with dimensions p×q, verify n=p. Reshape or transpose matrices using TRANSPOSE() if necessary.
Troubleshooting Checklist
- 1.Verify that the number of columns in array1 equals the number of rows in array2 (fundamental matrix multiplication rule)
- 2.Confirm all cells in both arrays contain numeric values only—no text, spaces, or error values (#N/A, #DIV/0!, etc.)
- 3.Check that array references use absolute references ($A$1:$C$3) if copying formulas, or relative references are intentionally dynamic
- 4.In Excel 2019 and earlier, ensure the formula was confirmed with Ctrl+Shift+Enter to register as an array formula (look for curly braces in formula bar)
- 5.Validate that the result matrix dimensions are correct: m×n multiplied by n×p should produce m×p results
- 6.Test with small sample matrices first to confirm logic before applying MMULT to large datasets
Edge Cases
Multiplying a 1×n row vector by an n×1 column vector
Behavior: Returns a single scalar value (1×1 matrix), not an n×n matrix. This is the dot product operation.
Solution: This is correct behavior for dot products. If you need an n×n outer product instead, reverse the order: multiply n×1 by 1×n.
Common source of confusion—understand that 1×n × n×1 = 1×1, not n×n
One or both matrices contain zero values
Behavior: MMULT processes normally; zero values are valid numeric inputs and produce mathematically correct results with zero products.
Solution: No solution needed—this is expected behavior. Zeros propagate through calculations correctly.
Sparse matrices (mostly zeros) are handled efficiently; consider them for large-scale applications
Attempting to multiply non-square matrices with incompatible dimensions
Behavior: Returns #VALUE! error if column count of first matrix ≠ row count of second matrix.
Solution: Reshape matrices using TRANSPOSE, add/remove rows or columns, or reconsider your mathematical approach. Verify dimensions match: m×n must multiply n×p.
This is a safety feature—Excel prevents mathematically impossible operations
Limitations
- •MMULT only accepts two arrays; multiplying three or more matrices requires nested MMULT functions, which becomes complex and performance-intensive
- •Excel has practical size limits for arrays (approximately 16,384 columns and 1,048,576 rows); very large matrices may cause performance degradation or memory errors
- •MMULT cannot handle sparse matrices efficiently in native Excel; large matrices with mostly zeros should be processed in specialized software like Python or R
- •No built-in functionality for partial matrix multiplication or conditional operations within MMULT; complex logic requires alternative approaches or helper columns
Alternatives
Compatibility
✓ Excel
Since 2007
=MMULT(array1, array2) with Ctrl+Shift+Enter in 2007-2019; automatic in 365✓Google Sheets
=MMULT(array1, array2)Google Sheets treats MMULT as a native array formula without requiring Ctrl+Shift+Enter; results automatically spill to adjacent cells
✓LibreOffice
=MMULT(array1, array2) with Ctrl+Shift+Enter for array confirmation