{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2026-05-26T00:16:20.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2026-05-26T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":1051,"title":"Exact binary matrix factorization","description":"Given a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u003cM and K\u003cN) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\r\n\r\n  C =\r\n  \r\n       1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1\r\n\r\nthen one possible solution is\r\n\r\n  A =\r\n\r\n     0     1\r\n     0     0\r\n     1     0\r\n  \r\n  B =\r\n  \r\n       0     1     0     1\r\n       1     1     1     0\r\n\r\nFor this problem, assume that at least one solution exists.\r\n\r\nNote that you cannot do matrix multiplication with two logical matrices in MATLAB. ","description_html":"\u003cp\u003eGiven a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u0026lt;M and K\u0026lt;N) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eC =\r\n\u003c/pre\u003e\u003cpre\u003e       1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1\u003c/pre\u003e\u003cp\u003ethen one possible solution is\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eA =\r\n\u003c/pre\u003e\u003cpre\u003e     0     1\r\n     0     0\r\n     1     0\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eB =\r\n\u003c/pre\u003e\u003cpre\u003e       0     1     0     1\r\n       1     1     1     0\u003c/pre\u003e\u003cp\u003eFor this problem, assume that at least one solution exists.\u003c/p\u003e\u003cp\u003eNote that you cannot do matrix multiplication with two logical matrices in MATLAB.\u003c/p\u003e","function_template":"function [A,B] = ebmf(C,K)\r\n  A = ones(size(C,1),K);\r\n  B = ones(K,size(C,2));\r\nend","test_suite":"%%\r\nC =   [1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%%\r\nC = [1     0     0\r\n     1     0     0\r\n     1     1     0];\r\nK = 3;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%%\r\nC = [0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0];\r\nK = 1;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%% Added another small test case\r\nC = [1     0     1\r\n     1     1     1\r\n     0     1     0];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%% A degenerate problem\r\nC = [0     0     1\r\n     0     0     0\r\n     0     0     1];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":450,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":"2012-11-26T07:45:37.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-11-23T20:46:50.000Z","updated_at":"2026-05-26T13:45:29.000Z","published_at":"2012-11-23T21:13:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u0026lt;M and K\u0026lt;N) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[C =\\n\\n       1     1     1     0\\n       0     0     0     0\\n       0     1     0     1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethen one possible solution is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[A =\\n\\n     0     1\\n     0     0\\n     1     0\\n\\nB =\\n\\n       0     1     0     1\\n       1     1     1     0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, assume that at least one solution exists.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote that you cannot do matrix multiplication with two logical matrices in MATLAB.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"problems":[{"id":1051,"title":"Exact binary matrix factorization","description":"Given a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u003cM and K\u003cN) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\r\n\r\n  C =\r\n  \r\n       1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1\r\n\r\nthen one possible solution is\r\n\r\n  A =\r\n\r\n     0     1\r\n     0     0\r\n     1     0\r\n  \r\n  B =\r\n  \r\n       0     1     0     1\r\n       1     1     1     0\r\n\r\nFor this problem, assume that at least one solution exists.\r\n\r\nNote that you cannot do matrix multiplication with two logical matrices in MATLAB. ","description_html":"\u003cp\u003eGiven a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u0026lt;M and K\u0026lt;N) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eC =\r\n\u003c/pre\u003e\u003cpre\u003e       1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1\u003c/pre\u003e\u003cp\u003ethen one possible solution is\u003c/p\u003e\u003cpre class=\"language-matlab\"\u003eA =\r\n\u003c/pre\u003e\u003cpre\u003e     0     1\r\n     0     0\r\n     1     0\u003c/pre\u003e\u003cpre class=\"language-matlab\"\u003eB =\r\n\u003c/pre\u003e\u003cpre\u003e       0     1     0     1\r\n       1     1     1     0\u003c/pre\u003e\u003cp\u003eFor this problem, assume that at least one solution exists.\u003c/p\u003e\u003cp\u003eNote that you cannot do matrix multiplication with two logical matrices in MATLAB.\u003c/p\u003e","function_template":"function [A,B] = ebmf(C,K)\r\n  A = ones(size(C,1),K);\r\n  B = ones(K,size(C,2));\r\nend","test_suite":"%%\r\nC =   [1     1     1     0\r\n       0     0     0     0\r\n       0     1     0     1];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%%\r\nC = [1     0     0\r\n     1     0     0\r\n     1     1     0];\r\nK = 3;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%%\r\nC = [0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     1     0     0     0     1     1     1     0     1     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0\r\n     0     0     0     0     0     0     0     0     0     0];\r\nK = 1;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%% Added another small test case\r\nC = [1     0     1\r\n     1     1     1\r\n     0     1     0];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))\r\n\r\n%% A degenerate problem\r\nC = [0     0     1\r\n     0     0     0\r\n     0     0     1];\r\nK = 2;\r\n[A,B] = ebmf(C,K);\r\nassert(isequal(A*B,C))\r\nassert(isequal(unique(A),[0; 1]))\r\nassert(isequal(unique(B'),[0; 1]))\r\nassert(all(size(A)==[size(C,1),K]))\r\nassert(all(size(B)==[K,size(C,2)]))","published":true,"deleted":false,"likes_count":0,"comments_count":3,"created_by":450,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":15,"test_suite_updated_at":"2012-11-26T07:45:37.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2012-11-23T20:46:50.000Z","updated_at":"2026-05-26T13:45:29.000Z","published_at":"2012-11-23T21:13:28.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven a binary-valued (only ones and zeroes) MxN matrix C, and K (where K\u0026lt;M and K\u0026lt;N) find a binary-valued MxK matrix A and a binary-valued KxN matrix B such that A*B equals C. e.g. for K = 2, and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[C =\\n\\n       1     1     1     0\\n       0     0     0     0\\n       0     1     0     1]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003ethen one possible solution is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[A =\\n\\n     0     1\\n     0     0\\n     1     0\\n\\nB =\\n\\n       0     1     0     1\\n       1     1     1     0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor this problem, assume that at least one solution exists.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eNote that you cannot do matrix multiplication with two logical matrices in MATLAB.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"errors":[],"facets":[[],[{"value":"medium","count":1,"selected":false}]],"term":"tag:\"matrix factorization\"","page":1,"per_page":50,"sort":"map(difficulty_value,0,0,999) asc"}}