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Accedmychevron_right11thchevron_rightmathchevron_rightMatrices And Determinantschevron_rightExercise 4.3

Solution

The rank of a matrix equals the number of non-zero rows in its row-echelon (or reduced row-echelon) form.

(i) [1−131−2−61−1314−2]\begin{bmatrix}1&-1&3&1\\-2&-6&1&-1\\3&1&4&-2\end{bmatrix}

Row-reduction gives 3 pivots, so:

rank⁡=3\boxed{\operatorname{rank}=3}

(ii) [1−232−46−10201−1]\begin{bmatrix}1&-2&3\\2&-4&6\\-1&0&2\\0&1&-1\end{bmatrix}

Row-reduction gives 3 pivots, so:

rank⁡=3\boxed{\operatorname{rank}=3}

(iii) [3−130112−1−3−22342125−2−33]\begin{bmatrix}3&-1&3&0&1\\1&2&-1&-3&-2\\2&3&4&2&1\\2&5&-2&-3&3\end{bmatrix}

Row-reduction gives 4 pivots, so:

rank⁡=4\boxed{\operatorname{rank}=4}
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