## 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 .

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