Question

[M] Repeat Exercise 37 for three random-integer-valued \(5 \times 7\) matrices \(A\) whose ranks are 5, 4, and 3. Make a conjecture about how \(CR\) is related to \(A\) for any matrix \(A\). Prove your conjecture.

Short answer

The weights of \({r_j}\) are needed to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).

Step-by-step solution

Write the matrix as in Exercise 37

Consider matrix\(A\)as\(A = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&{ - 4}&5&3&{ - 3}&{ - 7}\\{ - 4}&6&7&{ - 2}&{ - 6}&{ - 5}&5\\5&{ - 7}&{ - 6}&5&{ - 6}&2&8\\{ - 3}&5&8&{ - 1}&{ - 7}&{ - 4}&8\\6&{ - 8}&{ - 5}&4&4&9&3\end{array}} \right]\).

Matrix \(C\) is \(C = \left[ {\begin{array}{*{20}{c}}7&{ - 9}&5&{ - 3}\\{ - 4}&6&{ - 2}&{ - 5}\\5&{ - 7}&5&2\\{ - 3}&5&{ - 1}&{ - 4}\\6&{ - 8}&4&9\end{array}} \right]\).

Matrix \(R\) is \(R = \left[ {\begin{array}{*{20}{c}}1&0&{\frac{{13}}{2}}&0&5&0&{ - 3}\\0&1&{\frac{{11}}{2}}&0&{\frac{1}{2}}&0&2\\0&0&0&1&{\frac{{ - 11}}{2}}&0&7\\0&0&0&0&0&1&1\end{array}} \right]\).

Make a conjecture about how \(CR\) is related to \(A\)

When \(A\) is nonzero, \(A = CR\). \(CR = \left[ {\begin{array}{*{20}{c}}{C{{\mathop{\rm r}\nolimits} _1}}&{C{{\mathop{\rm r}\nolimits} _2}}& \cdots &{C{{\mathop{\rm r}\nolimits} _n}}\end{array}} \right]\), where \({{\mathop{\rm r}\nolimits} _1},...,{{\mathop{\rm r}\nolimits} _n}\) represent the columns of \(R\). The columns of \(R\) may or may not have the pivot columns of \(R\). Let the first column of \(R\), \({e_i}\) be the \({i^{th}}\) pivot column of \(R\); the \({i^{th}}\) column in the identity matrix. Thus, \(C{e_i}\) is the \({i^{th}}\) pivot column of \(A\). Multiply \(C\) with the pivot column of \(R\) and the result is the corresponding pivot column of \(A\) in its pivot position because both \(A\) and \(R\) have the pivot columns in the same location.

Consider \({r_j}\) as a non-pivot column of \(R\). So, \({r_j}\) contains the weights needed to construct the \({j^{th}}\) column of \(A\) from the pivot columns of \(A\) as explained in examples 9 and 10 of section 4.3. Therefore, the weights of \({r_j}\) are used to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).

Thus, \({r_j}\) contains the weights needed to construct the \({j^{th}}\) column of \(A\) from the columns of \(C\) and \(C{{\mathop{\rm r}\nolimits} _j} = {{\mathop{\rm a}\nolimits} _j}\).