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Chapter 2: Q21Q (page 93)

Suppose the last column of ABis entirely zero but Bitself has no column of zeros. What can you sayaboutthe columns of A?

Short Answer

Expert verified

The columns of Aare linearly dependent.

Step by step solution

01

The last column of AB is zero

Consider \({b_p}\) to be the last column of B. The last column of \(AB\) should be zero, according to the hypothesis.

02

Determine the columns of A

Here, \(A{b_p} = 0\). On the other hand, Bcontains no column of zeros; therefore, \({b_p}\) is not the zero vector. The equation \(A{b_p} = 0\) represents the linear dependence relation among the columns of A.

Thus, the columns of Aare linearly dependent.

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Most popular questions from this chapter

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 \(AB = BA\).

Suppose the second column of Bis all zeros. What can you

say about the second column of AB?

Suppose the third column of Bis the sum of the first two columns. What can you say about the third column of AB? Why?

Suppose the first two columns, \({{\bf{b}}_1}\) and \({{\bf{b}}_2}\), of Bare equal. What can you say about the columns of AB(if ABis defined)? Why?

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

1. \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\E&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\)
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