// Numbas version: exam_results_page_options {"name": "John's copy of Row reducing a matrix and finding its rank and nullity-MA2223", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "deter<>0\nand\nmax(map(testing(record[0][x]),x,0..length(record[0])-1))=0", "maxRuns": "200"}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "notes": "

Rank can be 2,3 or 4

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Reduce a 5x6 matrix to row reduced form and using this find rank and nullity.

a) The following shows how \$A\$ is reduced to row-echelon form.

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{solution(record_ops_matrix,record_ops_message)}

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b)

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The rank of A is the number of columns of R with pivots, i.e. \$\\var{rank}\$.

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The nullity of A is the number of columns of R without pivots, i.e. \$5-\\var{rank}=\\var{5-rank}\$.

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Remember that the Rank-Nullity Theorem says that the number of columns of A equals the rank of A plus the nullity of A, i.e. \$5=\\var{rank}+\\var{nullity}\$.

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Also the dimensions of the row space and the column space are both equal to the rank.

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Finally, the dimension of the null space is the nullity.

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\$R= \$[[0]]

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All entries in the matrix must be input as fractions or integers and not as decimals.

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\$\\operatorname{Rank}(A)=\$[[0]]

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\$\\operatorname{Nullity}(A)=\$[[1]]

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\$\\operatorname{dim}(\\operatorname{col}(A))=\$[[2]]

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\$\\operatorname{dim}(\\operatorname{row}(A))=\$[[3]]

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\$\\operatorname{dim}(\\operatorname{null}(A))=\$[[4]]

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