Matrix square expansion repair with both cross products

Matrix square expansion repair begins by keeping the order of each product. In this original paper exercise, two supplied arrays each square to the zero array, yet their sum squares to the identity. The missing contribution comes from the two cross products. Both are defined, and they differ from one another.

The task is to preserve the original inputs while correcting two proposed expansions. One leaves out both cross products. The other replaces their sum with twice the same product. We can check each proposal using exact integers, then explain the condition under which the second shortcut would work. Nothing here is a measurement, device output or claim about a product performing arithmetic.

Write the two arrays and the operation

Let A have first row 0, 1 and second row 0, 0. Let B have first row 0, 0 and second row 1, 0. Each is a two-by-two array with exact real entries. The zeros in these descriptions are supplied values, not blanks. Keep row breaks visible on the worksheet so that a row cannot silently become a column during copying.

Define addition entry by entry. For multiplication, pair a whole left row with a whole right column, multiply corresponding entries, and add the two products. Thus the upper-left entry of AB uses the first row of A and first column of B. The first letter names the left operand. A square means multiplication of an array by itself under this rule, rather than squaring each entry separately.

Square the sum directly

Adding the two arrays gives S with first row 0, 1 and second row 1, 0. Its first column is 0, 1 and its second column is 1, 0. Keep S beside its two original summands. We have changed neither operand; S is the entrywise sum specified by the question.

Multiply S by itself. The upper-left calculation is 0 times 0 plus 1 times 1, giving 1. The upper-right calculation is 0 times 1 plus 1 times 0, giving 0. The second row similarly gives 0, 1. Therefore S squared has rows 1, 0 and 0, 1. Call that array I, the two-by-two identity. Its action on a column with entries u, v returns u, v.

Check the proposed separate squares

Now calculate A squared. The first column of A is entirely zero. Its second column is 1, 0. Pairing its first row 0, 1 with those columns gives 0 and 0. Its second row is entirely zero, so both remaining entries are zero too. A squared is the zero array Z, whose four entries are all zero.

B squared is also Z. Its first row is zero, giving two zero outputs. Its second row 1, 0 meets columns 0, 1 and 0, 0, again giving zero in both positions. Consequently A squared plus B squared equals Z. That cannot equal the directly computed I: their upper-left entries already disagree. Correct separate squares do not justify omitting the other products from the expansion.

Compute both ordered cross products

For AB, the first row of A is 0, 1 and the first column of B is 0, 1. Their paired sum is 1. The second column of B is zero, giving an upper-right entry of zero. A's zero second row gives two more zeros. AB therefore has rows 1, 0 and 0, 0. It retains the first coordinate of a column and sets the second to zero.

For BA, the first row of B is zero. Its two output entries are zero. The second row 1, 0 meets A's first column 0, 0, giving zero, and its second column 1, 0, giving one. Thus BA has rows 0, 0 and 0, 1. It retains the second coordinate instead. AB and BA have identical shapes, but their entries differ. Both are legitimate products of the original arrays.

Restore the complete expansion

Expand the two factors in (A + B)(A + B) without exchanging their order. Choosing A from the left factor and A from the right gives A squared. Choosing left A and right B gives AB. Choosing left B and right A gives BA. Choosing B twice gives B squared. Each choice has its own place, yielding A squared + AB + BA + B squared.

For our inputs, the outer two contributions are Z. The middle contributions have rows 1, 0; 0, 0 and 0, 0; 0, 1. Adding them entrywise gives I. The repaired expansion now agrees with the direct square. Keep the zero contributions written in the first corrected line, even if you later omit them for brevity. That makes clear why there are four terms before their actual values are substituted.

Explain why the four terms always appear

The expansion has an entrywise justification beyond this example. Fix an output row i and column j. Its value is the sum over the two index positions k of (a at i,k plus b at i,k) times (a at k,j plus b at k,j). These are ordinary real numbers inside each scalar multiplication. Expanding each scalar product produces four contributions for that k.

Collect those finite sums without changing the role of any entry. The a-times-a terms form A squared; a-times-b forms AB; b-times-a forms BA; b-times-b forms B squared. This establishes the four-term identity for arbitrary real two-by-two inputs under the defined operations. It does not require AB to equal BA. Ordinary scalar distributivity justifies the expansion while the index positions preserve which matrix product each sum represents.

Locate the extra assumption in twice AB

A second proposed answer keeps a cross term but writes A squared + 2AB + B squared. Here 2AB means multiplying every entry of AB by two. For our pair that gives rows 2, 0 and 0, 0. Adding the zero separate squares leaves that same array. Its lower-right entry is zero rather than the required one, so this proposal also fails the original calculation.

The correct cross contribution is AB + BA. It equals 2AB exactly when BA equals AB: subtract AB entrywise from both sides to obtain that condition, and add AB back to reverse the argument. This is an exact equivalence for these real arrays. Calling the products commuting means precisely that their two orders agree. The shortcut therefore needs an established equality of ordered products; matching dimensions alone cannot supply it.

Keep a successful shortcut in its own control

For a separate control, take C with diagonal entries 2 and 3 and off-diagonal entries zero. Take D with diagonal entries 5 and 7 and off-diagonal entries zero. CD and DC both have diagonal entries 10 and 21. Their cross products can consequently combine. C + D has diagonal entries 7 and 10, whose square has diagonal entries 49 and 100.

The shortcut gives the same result in that control. Its first diagonal calculation is 4 + 20 + 25 = 49; its second is 9 + 42 + 49 = 100. All off-diagonal entries remain zero. These changed inputs show a permitted use of the shortcut. They do not repair the original A and B by replacing their entries, and they do not establish that every two-by-two pair commutes.

The zero array supplies another simple control: pairing any array with Z makes both cross products zero. A finite exact check also examined every ordered pair of two-by-two arrays whose entries belong to minus one, zero and one. There are 81 arrays and 6,561 ordered pairs. All passed the four-term identity. Exactly 817 pairs commuted, and exactly those 817 passed the twice-AB shortcut. The entrywise proof explains the general result; this finite enumeration checks its stated range.

Check a scaled version with the same structure

Keep the original A and B, but make a separately labeled calculation with E = 2A and F = 3B. E has first row 0, 2 and second row 0, 0. F has first row 0, 0 and second row 3, 0. Their sum has rows 0, 2 and 3, 0. Direct multiplication gives diagonal entries six and six, with both off-diagonal entries zero.

E squared and F squared are still Z. EF has diagonal entries six and zero, while FE has diagonal entries zero and six. Together those ordered cross products supply the two diagonal sixes in the direct result. Twice EF would instead put twelve in the upper-left position and zero in the lower-right. The same two mistaken expansion patterns therefore remain visible after this explicit change of scale.

More generally, let E = aA and F = bB for declared real scalars a and b. Their sum has rows 0, a and b, 0. Squaring directly gives ab in each diagonal position and zero elsewhere. EF places ab only in the first diagonal position; FE places it only in the second. This calculation explains the scaled family from its entries, without assuming an additional matrix identity merely because the notation resembles scalar multiplication.

When a or b equals zero, ab equals zero and both cross products vanish. The shortcut then succeeds for that restricted input. When both scalars are nonzero, EF and FE differ and the shortcut fails. A check using only a zero-scaled example would therefore miss the defect. Retain the two-six result with a = 2 and b = 3 beside the general formula, and keep all these scaled records separate from the original unit-entry arrays.

Preserve the inputs with the diagnosis

Keep three results beside the original A and B: the direct sum square I, the incomplete separate-square result Z, and the unsupported twice-AB result with diagonal 2, 0. Name the missing operation in each proposed line. The first omits both ordered products. The second assumes they agree. This record lets another reader locate the mistake without changing an input or guessing which earlier arithmetic should be discarded.

The matrix chain operation cost guide compares two parenthesizations of three fixed ordered operands under a declared multiplication count. This exercise instead expands a sum of two arrays; it supplies no execution-time or operation-cost saving.

The matrix shape compatibility guide distinguishes a defined rectangular product from an undefined reverse order. Here AB and BA are both defined and both two by two. Their calculated entries determine whether the cross terms can combine.

Keep the actual weather-vane accounts bounded

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The yotijay owl weather-vane account has nine images showing an owl arrow, attached colored cup arms, direction letters and a three-prong foot. Garden backgrounds are staging. No supplied extra stock, wind-speed sensor, measurement scale, material, dimensions or operating result is verified. Its architecture differs from the rooster assembly beyond color or animal decoration.

Keep both ordered cross products beside the original arrays.

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