Rank-one matrix compatibility: fill only the forced entries

A blank cell needs a stated constraint

Rank-one matrix compatibility asks whether known entries can belong to an array whose rows all follow one common row pattern. A plausible number in a blank cell is not enough. The fixed entries must support the same constraint, and the constraint may force one completion, permit many, or permit none. This original exact arithmetic worksheet gives an example of each outcome. Its entries are declared real numbers, with no measurement noise or missing observations from an actual dataset.

Take a three by three array with first row 2, 4, 6; second row 3, 6, x; and third row 0, y, 0. The letters x and y mark unknown real entries. Every displayed numeral is fixed, including each zero. We want rank at most one. In this exercise that means all rows are scalar multiples of one common nonzero row, with the all-zero array also allowed. Since the fixed first row contains 2, the completed main array cannot be the all-zero array.

This condition supplies a specific meaning for compatibility. It does not mean that the array looks balanced or that its entries fit a familiar sequence. Nor does it ask whether two arrays can be multiplied. The task is to retain all the known entries and determine whether suitable values of x and y make the stated row relationship hold. Altering a fixed entry would produce another problem, even if that alteration made a pattern easier to complete.

Use the nonzero entry without guessing

The first row is nonzero, so it can serve as the common row. If the second row is its scalar multiple, its first entry must satisfy 3=k times 2. This fixes k=3/2. The second entry then has to be k times 4, which equals 6 and agrees with the known entry. The third entry must be k times 6, which equals 9. Thus x is forced to be 9 if any compatible completion exists.

The third row begins with zero. If it equals h times the first row, its first coordinate gives 0=h times 2. Because 2 is nonzero, h=0. Its middle entry must therefore be zero times 4, or zero, so y=0. Its last fixed entry is also zero, matching zero times 6. This argument uses the same row multiplier for every coordinate in a given row. Choosing a different multiplier for each coordinate would discard the rank constraint.

These steps prove necessity: every compatible completion must use x=9 and y=0. They do not yet justify accepting a completed array without checking it. A deduction can overlook another fixed entry that conflicts with the required multiplier. Here we checked the second row's known middle entry and the third row's known last entry explicitly. The next step verifies the entire proposed completion rather than treating the two solved equations as a complete certificate on their own.

Verify the completed array

Insert the two values. The rows become (2,4,6), (3,6,9), and (0,0,0). They are respectively 1, 3/2, and 0 times (2,4,6). Every coordinate agrees with those whole-row multipliers. This gives the required common-row representation, proving sufficiency. Because necessity forced both unknowns and sufficiency verifies them, the completion exists and is unique. Its rank is exactly one because it has a nonzero row and all rows belong to that row's one dimensional span.

A second certificate uses two by two cross-products. Choose two rows and two columns, retaining their order, and form a small block with entries a, b in the upper row and c, d in the lower row. Its determinant is ad-bc. For rows that share one common pattern, this expression is zero. In the main array, the block from the first two rows and first and third columns gives 2x-6 times 3=0, which forces x=9.

The first and third rows with the first two columns give 2y-4 times 0=0, forcing y=0. There are three choices of two rows and three choices of two columns in a three by three array, making nine such blocks. After completing the array, all nine determinants are zero. The row-multiple representation already proves the rank statement; these nine checks provide another explicit record. Do not substitute a check of only one convenient block for verifying all known entries.

The completed array also has a column description. Its columns are (2,3,0), (4,6,0), and (6,9,0). They are 1, 2, and 3 times the nonzero first column. This check retains the order of entries down a column, rather than reading a row as a column by accident. It reaches the same completed array from the other direction. Each unknown belongs to both a row and a column, so a claimed completion cannot use one value when checking rows and a different value when checking columns.

One compact representation uses the column vector (2,3,0) and the row vector (1,2,3). Multiply each entry of the column by every entry of the row in order. The first resulting row is 2 times (1,2,3), the second is 3 times that row vector, and the third is zero times it. This produces all nine entries of the completed array. These two explicitly given vectors provide a factorization certificate, not a second set of independently chosen values for the blanks.

Notice that the factors themselves need not be unique even though the completed entries are unique. Doubling the column to (4,6,0) and halving the row to (1/2,1,3/2) leaves every product unchanged. The same completed array therefore has at least these two factor descriptions. Distinguish uniqueness of the unknown entries from uniqueness of a chosen representation. The main problem asks the former question, and its necessary equations already settled it.

A fixed inconsistency prevents every completion

Consider a separate array whose rows are (2,4,6), (3,7,x), and (0,y,0). Only the known middle entry in the second row changed, from 6 to 7. The first coordinate still forces that row's multiplier to be 3/2. But multiplying the first row's middle entry 4 by 3/2 gives 6, not 7. No choice of x or y can remove this conflict between fixed entries.

The upper left two by two block makes the contradiction compact. Its determinant is 2 times 7 minus 4 times 3, or 14-12=2. That value is nonzero and contains neither unknown. A compatible rank-at-most-one array would have to make the same determinant zero, so no completion exists. Repeatedly trying values for the blank cells would miss the cause. The failure is already located in a block of four entries that the problem does not permit changing.

Keep this impossible case separate from the successful case. Replacing the 7 with 6 would restore the earlier data, but it would amend an input. It would not complete the array while preserving every fixed entry. A useful record therefore identifies both the changed input and the fixed conflicting block. That distinction lets another reader tell whether a proposed answer solves the assigned problem or solves a quietly revised version of it.

A zero chosen pivot does not settle the answer

The first example used a nonzero entry as a pivot. A zero in that chosen position would make division by that entry unavailable, but it would not by itself prove that a completion is impossible. To see why, take a two by two array with rows (0,0) and (3,x). For every real x, the first row is zero and the second row is nonzero because its first entry is 3. Its rank is exactly one for every real choice of x.

Compare the separate array with rows (0,1) and (1,x). Its determinant is zero times x minus 1 times 1, which equals -1 for every real x. It has rank two and cannot satisfy the rank-at-most-one constraint. Both arrays have a zero in the upper left position. One permits every real x, while the other permits none. The remaining known entries, rather than the zero pivot alone, distinguish these outcomes.

For a general two by two array with rows (a,b) and (c,x), the rank-at-most-one condition is ax=bc. If the known a is nonzero, x=bc/a is the unique solution. If a=0 and bc is nonzero, the equation has no solution. If a=0 and bc=0, every real x satisfies it. This classification concerns rank at most one; deciding whether a completed array has rank zero or exactly one additionally requires checking whether all its entries are zero.

The general condition can be checked directly without a division by a zero entry. For nonzero a, the second row must be c/a times the first row, giving the stated equation and construction. When a=0, a nonzero b together with a nonzero c produces the impossible off-diagonal block. If either b or c is zero, one row or one column can be zero, and the remaining row or column supplies a one dimensional representation for any x. The all-zero case remains allowed.

Check the scope of the calculation

An independent exact arithmetic control used rational Gaussian elimination to calculate rank. It checked 875 two by two cases with a, b, c from -2 through 2 and x from -3 through 3. In every tested case, rank at most one agreed with ax=bc. A separate search over x and y from -3 through 10 found only (9,0) for the main array and no solution for the changed-7 array. These finite checks inspect stated inputs; the algebra and row representations supply the general reasoning.

The worksheet does not infer an approximately low-rank model from noisy data, estimate an error, or choose a tolerance. All known entries are exact. A tiny nonzero determinant and an exact zero would be different findings under this rule. Rounding a computed number and then calling it zero would introduce a new decision that the exercise has not defined. Likewise, replacing the row relationship with a visual trend would change the constraint rather than verify it.

Bound the optional native accounts

The account for Ashland iridescent tinsel garland has three gallery views of a decorated strand with attached spheres, dark decorative pieces, and stars. The catalog length is not an image measurement. Independent loose ornaments, maker identity, material, dimensions, delivered contents, and performance are not authenticated.

The account for Beistle blue tinsel garland 31561627 shows a blue strand and an inset detail. That inset does not establish another supplied strand. Catalog identifiers remain labels; material, dimensions, delivered quantity, maker authenticity, and performance are unverified. Neither account supplies the matrix or completion rule, and no purchase is needed.

Keep adjacent matrix questions separate

The matrix shape compatibility guide checks whether complete row-column pairing defines a requested product. Here the array shape is fixed and no product of two arrays is requested. The unknown entries must meet a rank constraint rather than repair a mismatch in inner dimensions.

The inner product compatibility checks guide tests positivity for a proposed measurement rule. This exercise instead seeks a singular rank-one array and allows nonsymmetric entries. A valid completion here does not certify a positive-definite inner product. Retain the actual mathematical question beside the result.

Separate any transaction from the worksheet

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