{
  "index": "1981-B-4",
  "type": "ALG",
  "tag": [
    "ALG"
  ],
  "difficulty": "",
  "question": "Problem B-4\nLet \\( V \\) be a set of 5 by 7 matrices, with real entries and with the property that \\( r A+s B \\in V \\) whenever \\( A, B \\in V \\) and \\( r \\) and \\( s \\) are scalars (i.e., real numbers). Prove or disprove the following assertion: If \\( V \\) contains matrices of ranks 0,1 . 2,4 , and 5 , then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix \\( M \\) is the largest \\( k \\) such that the entries of some \\( k \\) rows and some \\( k \\) columns form a \\( k \\) by \\( k \\) matrix with a nonzero determinant.]",
  "solution": "B-4.\nLet \\( M=M(a, b, c) \\) denote the 5 by 7 matrix ( \\( a_{i j} \\) ) with\n\\[\na_{11}=a, a_{22}=a_{33}=a_{44}=a_{55}=b, a_{16}=a_{27}=c\n\\]\nand \\( a_{1},=0 \\) in all other cases. Then the set \\( V \\) of all such \\( M \\) (with \\( a, b, c \\) arbitrary real numbers) is closed under linear combinations. Also, \\( M(0,0,0), M(1,0,0), M(0,0,1), M(0,1,0) \\), and \\( M(1,1,0) \\) have ranks \\( 0,1,2,4 \\), and 5 , respectively. But no \\( M \\) in \\( V \\) has rank 3 since \\( b \\neq 0 \\) implies that the rank is 4 or 5 and \\( b=0 \\) forces the rank to be 0,1 , or 2 .",
  "vars": [
    "V",
    "A",
    "B",
    "r",
    "s",
    "M",
    "k",
    "a",
    "b",
    "c",
    "a_ij",
    "a_11",
    "a_22",
    "a_33",
    "a_44",
    "a_55",
    "a_16",
    "a_27",
    "i",
    "j"
  ],
  "params": [],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "V": "vectorspace",
        "A": "matrixa",
        "B": "matrixb",
        "r": "scalarone",
        "s": "scalartwo",
        "M": "matrixm",
        "k": "ranknum",
        "a": "coeffa",
        "b": "coeffb",
        "c": "coeffc",
        "a_ij": "entryij",
        "a_11": "entry11",
        "a_22": "entry22",
        "a_33": "entry33",
        "a_44": "entry44",
        "a_55": "entry55",
        "a_16": "entry16",
        "a_27": "entry27",
        "i": "indexi",
        "j": "indexj"
      },
      "question": "Problem B-4\nLet \\( vectorspace \\) be a set of 5 by 7 matrices, with real entries and with the property that \\( scalarone matrixa+scalartwo matrixb \\in vectorspace \\) whenever \\( matrixa, matrixb \\in vectorspace \\) and \\( scalarone \\) and \\( scalartwo \\) are scalars (i.e., real numbers). Prove or disprove the following assertion: If \\( vectorspace \\) contains matrices of ranks 0, 1, 2, 4, and 5, then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix \\( matrixm \\) is the largest \\( ranknum \\) such that the entries of some \\( ranknum \\) rows and some \\( ranknum \\) columns form a \\( ranknum \\) by \\( ranknum \\) matrix with a nonzero determinant.]",
      "solution": "B-4.\nLet \\( matrixm=matrixm(coeffa, coeffb, coeffc) \\) denote the 5 by 7 matrix \\( ( entryij ) \\) with\n\\[\nentry11=coeffa,\\; entry22=entry33=entry44=entry55=coeffb,\\; entry16=entry27=coeffc\n\\]\nand \\( entryij=0 \\) in all other cases. Then the set \\( vectorspace \\) of all such \\( matrixm \\) (with \\( coeffa, coeffb, coeffc \\) arbitrary real numbers) is closed under linear combinations. Also, \\( matrixm(0,0,0), matrixm(1,0,0), matrixm(0,0,1), matrixm(0,1,0) \\), and \\( matrixm(1,1,0) \\) have ranks \\( 0, 1, 2, 4 \\), and 5, respectively. But no \\( matrixm \\) in \\( vectorspace \\) has rank 3 since \\( coeffb \\neq 0 \\) implies that the rank is 4 or 5 and \\( coeffb=0 \\) forces the rank to be 0, 1, or 2."
    },
    "descriptive_long_confusing": {
      "map": {
        "V": "cantaloupe",
        "A": "turnipseed",
        "B": "dandelion",
        "r": "marblecake",
        "s": "tortoise",
        "M": "butterscotch",
        "k": "quesadilla",
        "a": "harmonica",
        "b": "chrysanthemum",
        "c": "zeppelin",
        "a_ij": "butterfinger",
        "a_11": "heffalump",
        "a_22": "jacksonion",
        "a_33": "peppercorn",
        "a_44": "thumbtack",
        "a_55": "hairbrush",
        "a_16": "vestigial",
        "a_27": "flibbertigibbet",
        "i": "xylophone",
        "j": "snowblower"
      },
      "question": "Problem B-4\nLet \\( cantaloupe \\) be a set of 5 by 7 matrices, with real entries and with the property that \\( marblecake\\, turnipseed + tortoise\\, dandelion \\in cantaloupe \\) whenever \\( turnipseed, dandelion \\in cantaloupe \\) and \\( marblecake \\) and \\( tortoise \\) are scalars (i.e., real numbers). Prove or disprove the following assertion: If \\( cantaloupe \\) contains matrices of ranks 0,1 . 2,4 , and 5 , then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix \\( butterscotch \\) is the largest \\( quesadilla \\) such that the entries of some \\( quesadilla \\) rows and some \\( quesadilla \\) columns form a \\( quesadilla \\) by \\( quesadilla \\) matrix with a nonzero determinant.]",
      "solution": "B-4.\nLet \\( butterscotch = butterscotch(harmonica, chrysanthemum, zeppelin) \\) denote the 5 by 7 matrix ( \\( butterfinger \\) ) with\n\\[\nheffalump = harmonica,\\; jacksonion = peppercorn = thumbtack = hairbrush = chrysanthemum,\\; vestigial = flibbertigibbet = zeppelin\n\\]\nand \\( butterfinger = 0 \\) in all other cases. Then the set \\( cantaloupe \\) of all such \\( butterscotch \\) (with \\( harmonica, chrysanthemum, zeppelin \\) arbitrary real numbers) is closed under linear combinations. Also, \\( butterscotch(0,0,0), butterscotch(1,0,0), butterscotch(0,0,1), butterscotch(0,1,0) \\), and \\( butterscotch(1,1,0) \\) have ranks \\( 0,1,2,4 \\), and 5 , respectively. But no \\( butterscotch \\) in \\( cantaloupe \\) has rank 3 since \\( chrysanthemum \\neq 0 \\) implies that the rank is 4 or 5 and \\( chrysanthemum = 0 \\) forces the rank to be 0,1 , or 2 ."
    },
    "descriptive_long_misleading": {
      "map": {
        "V": "scalarrealm",
        "A": "nullentity",
        "B": "voidentity",
        "r": "vectoritem",
        "s": "matrixitem",
        "M": "antimatrix",
        "k": "minicount",
        "a": "blankvalue",
        "b": "hollowvalue",
        "c": "emptyvalue",
        "a_ij": "staticentry",
        "a_11": "offcenteroneone",
        "a_22": "offcentertwotwo",
        "a_33": "offcenterthreethree",
        "a_44": "offcenterfourfour",
        "a_55": "offcenterfivefive",
        "a_16": "offcenteronesix",
        "a_27": "offcentertwoseven",
        "i": "constantindex",
        "j": "fixedindex"
      },
      "question": "Problem B-4\nLet \\( scalarrealm \\) be a set of 5 by 7 matrices, with real entries and with the property that \\( vectoritem nullentity+matrixitem voidentity \\in scalarrealm \\) whenever \\( nullentity, voidentity \\in scalarrealm \\) and \\( vectoritem \\) and \\( matrixitem \\) are scalars (i.e., real numbers). Prove or disprove the following assertion: If \\( scalarrealm \\) contains matrices of ranks 0,1,2,4, and 5, then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix \\( antimatrix \\) is the largest \\( minicount \\) such that the entries of some \\( minicount \\) rows and some \\( minicount \\) columns form a \\( minicount \\) by \\( minicount \\) matrix with a nonzero determinant.]",
      "solution": "B-4.\nLet \\( antimatrix=antimatrix(blankvalue, hollowvalue, emptyvalue) \\) denote the 5 by 7 matrix ( \\( staticentry \\) ) with\n\\[\noffcenteroneone=blankvalue,\\; offcentertwotwo=offcenterthreethree=offcenterfourfour=offcenterfivefive=hollowvalue,\\; offcenteronesix=offcentertwoseven=emptyvalue\n\\]\nand \\( staticentry,=0 \\) in all other cases. Then the set scalarrealm of all such antimatrix (with blankvalue, hollowvalue, emptyvalue arbitrary real numbers) is closed under linear combinations. Also, \\( antimatrix(0,0,0), antimatrix(1,0,0), antimatrix(0,0,1), antimatrix(0,1,0) \\), and \\( antimatrix(1,1,0) \\) have ranks \\( 0,1,2,4 \\), and 5 , respectively. But no antimatrix in scalarrealm has rank 3 since hollowvalue \\( \\neq 0 \\) implies that the rank is 4 or 5 and \\( hollowvalue=0 \\) forces the rank to be 0,1, or 2 ."
    },
    "garbled_string": {
      "map": {
        "V": "qzxwvtnp",
        "A": "hjgrksla",
        "B": "vnbdqkse",
        "r": "zljpward",
        "s": "xycptfgh",
        "M": "bdvkrgia",
        "k": "mdwqlsne",
        "a": "fltmxzie",
        "b": "tcrgohjx",
        "c": "vlsqipne",
        "a_ij": "kzugbrna",
        "a_11": "wnfctzsp",
        "a_22": "lrpvmaeg",
        "a_33": "ocjyrdkw",
        "a_44": "hbzqmfno",
        "a_55": "dvlkxwpr",
        "a_16": "spqldzve",
        "a_27": "nyhkxtgc",
        "i": "gzlhnqpe",
        "j": "mrxdvsko"
      },
      "question": "Problem B-4\nLet \\( qzxwvtnp \\) be a set of 5 by 7 matrices, with real entries and with the property that \\( zljpward hjgrksla+xycptfgh vnbdqkse \\in qzxwvtnp \\) whenever \\( hjgrksla, vnbdqkse \\in qzxwvtnp \\) and \\( zljpward \\) and \\( xycptfgh \\) are scalars (i.e., real numbers). Prove or disprove the following assertion: If \\( qzxwvtnp \\) contains matrices of ranks 0,1 . 2,4 , and 5 , then it also contains a matrix of rank 3.\n[The rank of a nonzero matrix \\( bdvkrgia \\) is the largest \\( mdwqlsne \\) such that the entries of some \\( mdwqlsne \\) rows and some \\( mdwqlsne \\) columns form a \\( mdwqlsne \\) by \\( mdwqlsne \\) matrix with a nonzero determinant.]",
      "solution": "B-4.\nLet \\( bdvkrgia=bdvkrgia(fltmxzie, tcrgohjx, vlsqipne) \\) denote the 5 by 7 matrix ( \\( kzugbrna \\) ) with\n\\[\nwnfctzsp=fltmxzie, lrpvmaeg=ocjyrdkw=hbzqmfno=dvlkxwpr=tcrgohjx, spqldzve=nyhkxtgc=vlsqipne\n\\]\nand \\( fltmxzie_{1},=0 \\) in all other cases. Then the set \\( qzxwvtnp \\) of all such \\( bdvkrgia \\) (with \\( fltmxzie, tcrgohjx, vlsqipne \\) arbitrary real numbers) is closed under linear combinations. Also, \\( bdvkrgia(0,0,0), bdvkrgia(1,0,0), bdvkrgia(0,0,1), bdvkrgia(0,1,0) \\), and \\( bdvkrgia(1,1,0) \\) have ranks \\( 0,1,2,4 \\), and 5 , respectively. But no \\( bdvkrgia \\) in \\( qzxwvtnp \\) has rank 3 since \\( tcrgohjx \\neq 0 \\) implies that the rank is 4 or 5 and \\( tcrgohjx=0 \\) forces the rank to be 0,1 , or 2 ."
    },
    "kernel_variant": {
      "question": "Let V be a collection of 6\\times 8 real matrices with the property that whenever A,B\\in V and r,s\\in\\mathbb R, the linear combination rA+sB also lies in V.  Suppose that V contains matrices whose ranks are 0,1,2,4, and 5.  Must V necessarily contain a matrix of rank 3?  Prove your answer.",
      "solution": "Answer: No. Here is a corrected version of the usual 3-parameter counterexample.\n\n1. Definition of the subspace V\nFor (a,b,c) in R^3, let M(a,b,c) be the 6×8 real matrix whose only nonzero entries are\n  M(1,1)=M(3,3)=M(4,4)=M(6,6)=b,\n  M(2,2)=a,\n  M(2,7)=M(5,8)=c.\nAll other entries are zero.  Clearly (a,b,c)→M(a,b,c) is linear, so\n  V={M(a,b,c): a,b,c in R}\nis a 3-dimensional subspace of the space of all 6×8 matrices.\n\n2. Exhibiting ranks 0,1,2,4,5 in V\n • Rank 0:  M(0,0,0)=0.\n • Rank 1:  M(2,0,0) has a single nonzero entry at (2,2), so rank 1.\n • Rank 2:  M(0,0,3) has nonzero entries at (2,7) and (5,8) in distinct rows and distinct columns, so rank 2.\n • Rank 4:  M(0,5,0) has the 4×4 block 5·I on rows {1,3,4,6} and columns {1,3,4,6}, giving rank 4.\n • Rank 5:  M(2,5,0) has that same 4×4 block (with b=5) plus the extra nonzero a=2 at (2,2), giving rank 5.\n\nThus V indeed contains matrices of ranks 0,1,2,4,5.\n\n3. No matrix in V can have rank 3\nWrite M=M(a,b,c).  Two cases:\n\nCase i) b=0.  Then M has nonzeros only in rows 2 and 5, so rank M ≤ 2.\n\nCase ii) b≠0.  The 4×4 submatrix on rows {1,3,4,6} and columns {1,3,4,6} is b·I₄, so rank M ≥ 4.  Moreover:\n  – If c≠0, then rows 2 and 5 each have a nonzero in a column disjoint from the first four, forcing rank M ≥ 6 (in fact rank 6).\n  – If c=0 but a≠0, then row 2 adds one new independent row (at column 2) so rank M = 5.\n  – If c=0 and a=0, then rank M = 4.\n\nIn no case can rank M = 3.  Therefore V contains no matrix of rank 3, even though it has ranks 0,1,2,4,5.\n\nConclusion: The answer to “Must V necessarily contain a matrix of rank 3?” is No. The 3-parameter family M(a,b,c) above is a subspace of 6×8 matrices that realizes exactly the ranks 0,1,2,4,5 and omits rank 3.",
      "_meta": {
        "core_steps": [
          "Define a 3-parameter template matrix M(a,b,c) and let V be the set of all such matrices (hence a vector space).",
          "Pick concrete parameter triples to obtain rank-0, rank-1, rank-2, rank-4, and rank-5 matrices inside V.",
          "Show that when b = 0, the rank of M(a,b,c) is at most 2, giving exactly the ranks 0,1,2 already exhibited.",
          "Show that when b ≠ 0, the rank of M(a,b,c) is at least 4 and can be 4 or 5, but never 3.",
          "Conclude that V contains the required ranks yet omits rank 3, so the original assertion is false."
        ],
        "mutable_slots": {
          "slot1": {
            "description": "Overall size of the matrices, provided it allows ranks up to 5 (e.g., any m×n with m ≥ 5 and n ≥ 7).",
            "original": "5×7"
          },
          "slot2": {
            "description": "Choice of the four diagonal positions that carry the common entry b (any four distinct diagonal indices would work).",
            "original": "{(2,2), (3,3), (4,4), (5,5)}"
          },
          "slot3": {
            "description": "Positions of the two off-diagonal entries that carry c; they only need to lie outside the chosen diagonal block and in distinct columns.",
            "original": "{(1,6), (2,7)}"
          },
          "slot4": {
            "description": "Specific non-zero parameter values used to produce the example ranks (e.g., 1 could be replaced by any non-zero scalar).",
            "original": "Triples (1,0,0), (0,0,1), (0,1,0), (1,1,0)"
          }
        }
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}