Find the rank of A and write the matrix as A=u v^T : A=[ 1 0 0 3 0 0

step 1 Hello and welcome to problem 2 .4 .12. We were asked to find the rank of A, write the matrices m

Find the rank of A and write the matrix as A=u v^T : A=[ ...

Find the rank of $A$ and write the matrix as $A=u v^{\mathrm{T}}$ :
$$
A=\left[\begin{array}{llll}
1 & 0 & 0 & 3 \\
0 & 0 & 0 & 0 \\
2 & 0 & 0 & 6
\end{array}\right] \quad \text { and } \quad A=\left[\begin{array}{ll}
2 & -2 \\
6 & -6
\end{array}\right]
$$

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Key Concepts

Outer Product Representation

The outer product representation of a matrix involves writing the matrix as the product of a column vector and a row vector. This method is applicable when the matrix has rank 1, meaning all its columns (or rows) are proportional. By finding appropriate vectors u and v, one can express the matrix succinctly, which can greatly simplify analysis and computations related to the matrix.

Rank-1 Matrix Decomposition

A rank-1 matrix can be expressed as the outer product of two vectors, meaning it can be written in the form A = u v^T. This decomposition explicitly shows the dependency structure of the matrix, as every row is a scalar multiple of the first row (or every column is a scalar multiple of the first column, depending on the context). This concept is particularly useful in simplifying matrix computations and understanding the structure of the matrix.

Matrix Rank

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It provides insight into the dimension of the space spanned by the matrix's rows or columns and indicates the level of redundancy among the vectors. In problems like these, determining the rank is crucial because it indicates whether the matrix is full rank or has dependencies making it possible to express it as an outer product if the rank is 1.

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