## Linear Algebra and Its Applications, Review Exercise 1.26

Review exercise 1.26. (a) Given a 3 by 3 matrix $A$ what vector $x$ would make the product $Ax$ have 1 times column 1 of A plus 2 times column 3?

(b) Construct a matrix $A$ for which the sum of column 1 and 2 times column 3 is zero, and show that $A$ is singular. Why? $x = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}$

then $Ax$ will consist of column 1 of $A$ plus 2 times column 3. In essence if $A = \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{bmatrix}$

then $Ax = \begin{bmatrix} a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23} \\ a_{31}&a_{32}&a_{33} \end{bmatrix} \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix} = 1 \cdot \begin{bmatrix} a_{11} \\ a_{21} \\ a_{31} \end{bmatrix} + 0 \cdot \begin{bmatrix} a_{12} \\ a_{22} \\ a_{32} \end{bmatrix} + 2 \cdot \begin{bmatrix} a_{13} \\ a_{23} \\ a_{33} \end{bmatrix}$

(b) The following matrix has column 1 plus 2 times column 3 equal to 0. $A = \begin{bmatrix} 2&1&-1 \\ -4&2&2 \\ 6&0&-3 \end{bmatrix}$

Elimination of this matrix proceeds as follows: $\begin{bmatrix} 2&1&-1 \\ -4&2&2 \\ 6&0&-3 \end{bmatrix} \rightarrow \begin{bmatrix} 2&1&-3 \\ 0&4&0 \\ 0&-3&0 \end{bmatrix} \rightarrow \begin{bmatrix} 2&1&-3 \\ 0&4&0 \\ 0&0&0 \end{bmatrix}$

Since there is no pivot in row 3 the matrix is singular. The problem is that since column 3 is a multiple of column 1 (being equal to -2 times column 1) the elimination steps that produce zeros in the $(2, 1)$ and $(3, 1)$ position will also produce zeros in the $(2, 3)$ and $(3, 3)$ positions, so that there is no possible pivot in column 3.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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