Science:Math Exam Resources/Courses/MATH221/April 2013/Question 02 (a)

MATH221 April 2013

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[hide] Question 02 (a)
Let $A=\left[{\begin{array}{ccccc}1&2&1&3&2\\3&4&9&0&7\\2&3&5&1&8\\2&2&8&-3&5\end{array}}\right]B=\left[{\begin{array}{ccccc}1&2&1&3&2\\0&1&-3&5&-4\\0&0&0&1&-7\\0&0&0&0&0\end{array}}\right]$ The matrices A and B are related by $B=MA$ , where M is an invertible 4x4 matrix. a) Find the rank and a basis of the row space of A.

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[show] Hint
How does left multiplication of a matrix by a invertible matrix change the rank and row space? What are properties preserved under those changes?

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[show] Solution

Since the matrix A and B are related by the left multiplication of that invertible matrix M, they share many characteristic, including rank and they row space. The reason being that the left multiplication by M can be understood as simply being a set of (invertible) row operations and the rank and row space are properties that are preserved under (invertible) row operations. Now, as B is in row echelon form, it is easier to read this information from the matrix B, so we will use to answer the question. As B has 3 pivots, we see that the rank is 3. We can also take each row that has a pivot to form a basis of the row space, thus our basis is given by: $\left[{\begin{array}{ccccc}1&2&1&3&2\\\end{array}}\right],\left[{\begin{array}{ccccc}0&1&-3&5&-4\\\end{array}}\right],\left[{\begin{array}{ccccc}0&0&0&1&-7\\\end{array}}\right]$

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