The matrix product ( x y z )( x y z ) (a) is not defined (b) equals (x^2+y^2

step 1 So in the given question we have been given a matrix product that is x, y, z times the matrix x,

The matrix product ( x y z )( x y z )...

The matrix product $\left(\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right)\left(\begin{array}{lll}\mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right)$ (a) is not defined (b) equals $\left(x^{2}+y^{2}+z^{2}\right)$ (c) equals $\left(\mathrm{x}^{2} \mathrm{y}^{2} \mathrm{z}^{2}\right)$ (d) is not invertible.

The matrix product $\left(\begin{array}{l}\mathrm{x} \\ \mathrm{y} \\ \mathrm{z}\end{array}\right)\left(\begin{array}{lll}\mathrm{x} & \mathrm{y} & \mathrm{z}\end{array}\right)$
(a) is not defined
(b) equals $\left(x^{2}+y^{2}+z^{2}\right)$
(c) equals $\left(\mathrm{x}^{2} \mathrm{y}^{2} \mathrm{z}^{2}\right)$
(d) is not invertible.

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Key Concepts

Matrix Rank

The rank of a matrix is the dimension of the vector space spanned by its rows or columns. In the case of an outer product of a non-zero vector with itself, the resultant matrix is of rank one because every row (or column) is a scalar multiple of a single vector. This limited rank means that the matrix does not have full rank when its dimensions are larger than one.

Invertibility

A square matrix is invertible if and only if its rank equals its size, meaning it has full rank. For a matrix formed by the outer product of a column vector and a row vector, the rank is typically one (unless the vector is the zero vector), which is less than the matrix's size when it is larger than a 1x1 matrix. Therefore, such matrices are not invertible due to their rank deficiency.

Outer Product

The outer product is the multiplication of a column vector by a row vector. In this operation, each entry in the resulting matrix is the product of an element from the column vector and an element from the row vector. This results in a matrix whose dimensions are determined by the lengths of the column and row vectors, as opposed to other products like the dot product, which results in a scalar.

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