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Chapter 2: Q2.8-35Q (page 93)
In Exercises 31-36, respond as comprehensively as possible, and justify your answer.
35. What can say about \({\mathop{\rm Nu}\nolimits} {\mathop{\rm l}\nolimits} \,\,B\) when B is a \(5 \times 4\) matrix with linearly independent columns?
Nul \(B = \left\{ 0 \right\}\).
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Suppose Ais an \(n \times n\) matrix with the property that the equation \(Ax = 0\)has only the trivial solution. Without using the Invertible Matrix Theorem, explain directly why the equation \(Ax = b\) must have a solution for each b in \({\mathbb{R}^n}\).
Prove the Theorem 3(d) i.e., \({\left( {AB} \right)^T} = {B^T}{A^T}\).
Let \(A = \left( {\begin{aligned}{*{20}{c}}1&1&1\\1&2&3\\1&4&5\end{aligned}} \right)\), and \(D = \left( {\begin{aligned}{*{20}{c}}2&0&0\\0&3&0\\0&0&5\end{aligned}} \right)\). Compute \(AD\) and \(DA\). Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a \(3 \times 3\) matrix B, not the identity matrix or the zero matrix, such that \(AB = BA\).
Let Abe an invertible \(n \times n\) matrix, and let B be an \(n \times p\) matrix. Show that the equation \(AX = B\) has a unique solution \({A^{ - 1}}B\).
Suppose the last column of ABis entirely zero but Bitself has no column of zeros. What can you sayaboutthe columns of A?
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