Let 𝐀=[ 3 0 4 -1 2 2 6 5 ...

Let $$\mathbf{A}=\left[\begin{array}{rrr}3 & 0 & 4 \\-1 & 2 & 2 \\6 & 5 & -4\end{array}\right], \mathbf{B}=\left[\begin{array}{rrr}0 & -5 & -3 \\-5 & 2 & 4 \\-3 & 4 & 0\end{array}\right],$$ $$\mathbf{C}=\left[\begin{array}{ll}0 & 2 \\2 & 4 \\1 & 3\end{array}\right], \quad \mathbf{D}=\left[\begin{array}{rr}6 & 1 \\-4 & 7 \\-8 & 3\end{array}\right],$$ $$\mathbf{u}=\left[\begin{array}{r}2 \\0 \\-1\end{array}\right], \quad\mathbf{v}=\left[\begin{array}{r}-45 \\0.8 \\1.2\end{array}\right].$$ Find the following expressions or give reasons why they are undefined. $$\mathbf{A}+\mathbf{u}, 12 \mathbf{a}+10 \mathbf{v}, 0(\mathbf{B}-\mathbf{v}), 0 \mathbf{B}+\mathbf{u}$$

Let $$\mathbf{A}=\left[\begin{array}{rrr}3 & 0 & 4 \\-1 & 2 & 2 \\6 & 5 & -4\end{array}\right], \mathbf{B}=\left[\begin{array}{rrr}0 & -5 & -3 \\-5 & 2 & 4 \\-3 & 4 & 0\end{array}\right],$$
$$\mathbf{C}=\left[\begin{array}{ll}0 & 2 \\2 & 4 \\1 & 3\end{array}\right], \quad \mathbf{D}=\left[\begin{array}{rr}6 & 1 \\-4 & 7 \\-8 & 3\end{array}\right],$$ $$\mathbf{u}=\left[\begin{array}{r}2 \\0 \\-1\end{array}\right], \quad\mathbf{v}=\left[\begin{array}{r}-45 \\0.8 \\1.2\end{array}\right].$$
Find the following expressions or give reasons why they are undefined.
$$\mathbf{A}+\mathbf{u}, 12 \mathbf{a}+10 \mathbf{v}, 0(\mathbf{B}-\mathbf{v}), 0 \mathbf{B}+\mathbf{u}$$

Key Concepts

Undefined Operations in Matrix Algebra

Not every algebraic expression involving matrices and vectors is meaningful. Operations such as adding a vector to a matrix of mismatched dimensions, or combining elements that are not of the same type (e.g., a vector and a matrix) are examples of undefined operations. Awareness of the required conditions for each operation helps in identifying when an expression is invalid.

Scalar Multiplication

Scalar multiplication involves multiplying every entry of a matrix or a vector by a constant (scalar). This operation is always well-defined regardless of the dimensions of the matrix or vector, and it is distributive over matrix addition, which makes it a fundamental operation in linear algebra.

Matrix and Vector Dimensions

This concept involves understanding that matrices and vectors have defined dimensions (number of rows and columns), and arithmetic operations such as addition and subtraction are only defined when the operands have exactly the same dimensions. If the dimensions don't match, the operation is undefined.

Matrix Addition and Subtraction

Matrix addition (or subtraction) requires that each corresponding entry in the matrices can be added together. This principle inherently depends on the matrices having the same shape. When an attempt is made to add a vector to a matrix or vice versa, one must first check if their dimensions align appropriately.

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