Apple Numbers MDETERM: Matrix Determinants and Worked Examples

• Introduced: Original release and introduction date are not established by the checked references.
• Availability: Currently documented in Apple’s Functions Help 15.4 documentation edition; no final supported-version endpoint is established.
• Scope: Shared iWork reference. Function-specific historical availability across Mac, iPad, iPhone and iCloud is unverified.
• Checked: October 14, 2026, against the official MDETERM reference. Independent exact matrix calculations; no native Numbers test.

=MDETERM(matrix)

The argument is a numeric array with the same number of rows and columns. Supply the data cells only. A header, label or extra summary column changes the input and may violate that contract. There are no optional arguments in the documented signature.

Apple explicitly states that unequal row and column counts produce an error. The page does not name the displayed error token. These examples use ordinary numeric cells, with English comma-separated formulas where a helper takes more than one argument.

Create a Numbers table named DeterminantCheck, with six columns A:F and five rows, including one header row. Put Coefficient 1 and Coefficient 2 in A1:B1. Enter the matrix:

In D2 enter =MDETERM(A2:B3); expect 5. In D3 enter =A2*B3-B2*A3; expect 5 again. D4 can contain =D2-D3, with expected result 0.

For a 2×2 matrix, the determinant is the product of the main diagonal minus the product of the other diagonal: 2×3 − 1×1. The explicit arithmetic makes an accidental selection or entry error easier to spot. The hand calculation is a mathematical check, not a description of Apple’s internal algorithm.

Create VendorDeterminant, with five columns and six rows. Use one header row and enter A2:C4:

The source’s matrix begins at A1; this worksheet moves its numbers down one row to leave room for headings. E3 selects only the top-left 2×2 block, so it answers a different question from E2. E4 expands along the first row: 5×11 − 3×16 + 7×39 = 280.

As a separate shape check, enter =MDETERM(A2:C3) in E6. This selection has two rows and three columns and is outside the documented square-matrix rule; expect an error, without relying on a particular error label. Keep E6 out of any later numeric summary.

Create MatrixControls, with eight columns and five rows. Enter three separate 2×2 blocks:

Put =MDETERM(A2:B3) in A5, =MDETERM(D2:E3) in D5 and =MDETERM(G2:H3) in G5. Expect −5, 50 and 0, respectively.

Exchanging two rows changes the sign. Scaling one row by ten scales the determinant by ten. The final matrix has dependent rows and zero determinant. In exact linear algebra, such a square matrix is singular and has no inverse. This mathematical conclusion is separate from which error an app might display for an attempted inverse.

Create ScaledMatrix, with four columns and five rows. Enter 0.002,0.001 in A2:B2 and 0.001,0.003 in A3:B3. In D2 enter =MDETERM(A2:B3); expect 0.000005. In D3 enter =D2/0.001^2; expect 5.

All four original coefficients were scaled by 0.001. For this 2×2 case, the determinant scales by its square, 0.000001. The result is small because of the units and scaling; its magnitude alone is not a universal measure of data quality or numerical reliability.

A displayed zero can also be a rounded nonzero value. Increase displayed precision before interpreting it, and do not assume an exact-equality test diagnoses every near-singular floating-point problem. The short reference does not establish native tolerances, numerical limits or every blank/text/error behavior.

MINVERSE returns an inverse matrix when one exists. MMULT multiplies matrices, while MUNIT builds an identity matrix. A determinant is useful for understanding invertibility, but it does not solve an entire system of equations by itself.