## Linear Algebra and Its Applications, Exercise 2.4.13

Exercise 2.4.13. What is the rank of each of the following matrices: $A = \begin{bmatrix} 1&0&0&3 \\ 0&0&0&0 \\ 2&0&0&6 \end{bmatrix} \qquad A = \begin{bmatrix} 2&-2 \\ 2&-2 \end{bmatrix}$

Express each matrix as a product of a column vector and row vector, $A = uv^T$.

Answer: We do Gaussian elimination on the first matrix by subtracting two times the first row from the third: $\begin{bmatrix} 1&0&0&3 \\ 0&0&0&0 \\ 2&0&0&6 \end{bmatrix} \Rightarrow \begin{bmatrix} 1&0&0&3 \\ 0&0&0&0 \\ 0&0&0&0 \end{bmatrix}$

The resulting echelon matrix has one pivot, and thus the rank of $A$ is 1. The matrix $A$ can be expressed as the product of a column vector and row vector as follows: $A = \begin{bmatrix} 1 \\ 0 \\ 2 \end{bmatrix}\begin{bmatrix} 1&0&0&3 \end{bmatrix}$

We do Gaussian elimination on the second matrix by subtracting the first row from the second: $\begin{bmatrix} 2&-2 \\ 2&-2 \end{bmatrix} \Rightarrow \begin{bmatrix} 2&-2 \\ 0&0 \end{bmatrix}$

The resulting echelon matrix has one pivot, and thus the rank of $A$ is 1. The matrix $A$ can be expressed as the product of a column vector and row vector as follows: $A = \begin{bmatrix} 2 \\ 2 \end{bmatrix} \begin{bmatrix} 1&-1 \end{bmatrix}$

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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