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Problem Set 1 15.If r is the rank and d is the determinant of the matrix what is r-d? Megan Grywalski

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Finding the Rank The matrix rank is determined by the number of independent rows or columns present in it. A row or a column is considered independent, if it satisfies the below conditions. 1. A row/column should have at least one non-zero element for it to be ranked. 2. A row/column should not be identical to another row/column. 3. A row/column should not be proportional (multiples) of another row/column. 4. A row/column should not be a linear combination of another row/column.

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Determining the Rank by putting the matrix in Row Echelon Form 1. The first element in the first row should be the leading element i.e. 1. 2. The leading element in the columns should be to the right of the previous row's leading element. 3. If there are any rows with all zero elements, it should be below the non-zero element rows. 4. The leading element should be the only non-zero element in every column.

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R 2 -4R 1 R 3 -7R 1 R 3 -2R 2 Since R 3 does not have at least one non-zero element it cannot be ranked. Therefore, the rank is 2.

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Finding the Determinant of a 3×3 Matrix = - + = 1(45-48) – 2(36-42) + 3(32-35) = (-3) + 12 – 9 = 0 So we have r = 2 and d = 0. Therefore, r-d = 2 – 0 = 2

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Section 1.7 Linear Independence and Nonsingular Matrices

Section 1.7 Linear Independence and Nonsingular Matrices

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