(a) 𝐀^T 𝐀=( 4 8 -10 )( 4 8 -10 )=(180) (b) 𝐁^T 𝐁=( 2 4

step 1 First part you have to find A transpose A transpose A transpose is B transpose is 4 8 minus 10 i

(a) 𝐀^T 𝐀=( 4 8 -10 )( 4 8 ...

(a) $\mathbf{A}^{T} \mathbf{A}=\left(\begin{array}{lll}4 & 8 & -10\end{array}\right)\left(\begin{array}{r}4 \\ 8 \\ -10\end{array}\right)=(180)$ (b) $\mathbf{B}^{T} \mathbf{B}=\left(\begin{array}{l}2 \\ 4 \\ 5\end{array}\right)\left(\begin{array}{lll}2 & 4 & 5\end{array}\right)=\left(\begin{array}{rrr}4 & 8 & 10 \\ 8 & 16 & 20 \\ 10 & 20 & 25\end{array}\right)$ (c) $\mathbf{A}+\mathbf{B}^{T}=\left(\begin{array}{r}4 \\ 8 \\ -10\end{array}\right)+\left(\begin{array}{l}2 \\ 4 \\ 5\end{array}\right)=\left(\begin{array}{r}6 \\ 12 \\ -5\end{array}\right)$

(a) $\mathbf{A}^{T} \mathbf{A}=\left(\begin{array}{lll}4 & 8 & -10\end{array}\right)\left(\begin{array}{r}4 \\ 8 \\ -10\end{array}\right)=(180)$
(b) $\mathbf{B}^{T} \mathbf{B}=\left(\begin{array}{l}2 \\ 4 \\ 5\end{array}\right)\left(\begin{array}{lll}2 & 4 & 5\end{array}\right)=\left(\begin{array}{rrr}4 & 8 & 10 \\ 8 & 16 & 20 \\ 10 & 20 & 25\end{array}\right)$
(c) $\mathbf{A}+\mathbf{B}^{T}=\left(\begin{array}{r}4 \\ 8 \\ -10\end{array}\right)+\left(\begin{array}{l}2 \\ 4 \\ 5\end{array}\right)=\left(\begin{array}{r}6 \\ 12 \\ -5\end{array}\right)$

Key Concepts

Outer Product

The outer product of two vectors, typically a column vector and a row vector, results in a matrix rather than a scalar. Each entry of the resulting matrix is the product of an element from the first vector and an element from the second vector. This operation is widely used in constructing matrices with particular properties and in tensor decomposition.

Matrix Addition

Matrix addition involves adding two matrices by summing their corresponding entries. For this operation to be valid, the matrices involved must have the same dimensions. It provides a method to combine matrices in a way that preserves their structure and is fundamental to many linear algebra operations.

Matrix Transpose

The matrix transpose is an operation that flips a matrix over its diagonal, interchanging its rows and columns. It is crucial in many matrix operations, such as forming inner products and outer products, because it allows a column vector to be transformed into a row vector (or vice versa) so that multiplication is possible according to the rules of linear algebra.

Inner Product (Dot Product)

The inner product, often called the dot product for vectors, is a way of multiplying two vectors to obtain a scalar. This operation involves multiplying corresponding entries and summing the results. It is an essential concept in determining lengths, angles, and testing orthogonality in vector spaces.

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