## Linear Algebra and Its Applications, Exercise 2.4.14

Exercise 2.4.14. Suppose we have the following matrix:

$A = \begin{bmatrix} a&b \\ c&d \end{bmatrix}$

with $a$, $b$, and $c$ given and $a \ne 0$. For what value of $d$ does $A$ have rank 1? In this case how can $A$ be expressed as the product of a column vector and row vector $A = uv^T$?

Answer:We can do Gaussian elimination on $A$ by multiplying the first row by $c/a$ (which is permissible since $a \ne 0$) and subtracting it from the second row:

$\begin{bmatrix} a&b \\ c&d \end{bmatrix} \Rightarrow \begin{bmatrix} a&b \\ 0&d-(c/a)b \end{bmatrix}$

Since $a \ne 0$ there is a pivot in the first column. For the rank of $A$ to be 1 that pivot must be the only one; for this to be true we must have $d - (c/a)b = 0$ or $d = (bc)/a$.

If $d = (bc)/a$ then the matrix $A$ can be expressed as the product of a column vector and row vector as follows:

$A = \begin{bmatrix} a&b \\ c&(bc)/a \end{bmatrix} = \begin{bmatrix} a \\ c \end{bmatrix} \begin{bmatrix} 1&b/a \end{bmatrix}$

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.

This entry was posted in linear algebra. Bookmark the permalink.