91Ó°ÊÓ

Understanding matrix dimensions is crucial when working with matrix operations. The dimensions of a matrix are given as rows x columns. In the original exercise, Matrix \( A \) is a column vector with dimensions \( 3 \times 1 \), meaning it has 3 rows and 1 column. Conversely, Matrix \( B \) is a row vector with dimensions \( 1 \times 3 \), having 1 row and 3 columns.

Knowing the dimensions of matrices allows you to determine the feasibility of matrix operations, such as multiplication and addition. For instance:

• Matrix multiplication \( AB \) is possible if the number of columns in \( A \) matches the number of rows in \( B \).
• Matrix addition or subtraction requires both matrices to have the same dimensions.

Clearly defining dimensions is the first step before performing operations to ensure all calculations are valid and correctly implementable.

Matrix scalar multiplication involves multiplying every element of a matrix by a scalar (a single number). It is a straightforward operation and is always defined, regardless of the matrix dimensions.

In the given exercise, two scalar multiplications are performed:

• For \( 3A \), each entry of Matrix \( A \) is multiplied by 3: \[3A = 3 \begin{bmatrix} 7 \ 8 \ 9 \end{bmatrix} = \begin{bmatrix} 21 \ 24 \ 27 \end{bmatrix}\]
• For \(-B\), each entry of Matrix \( B \) is multiplied by -1: \[-B = -1 \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} -1 & -2 & -3 \end{bmatrix}\]

This operation effectively scales the matrix by the given amount, affecting all elements in unison.

Matrix multiplication involves multiplying two matrices to produce a new matrix. It is not as straightforward as scalar multiplication because it requires specific conditions relating to the matrix dimensions.

Matrix multiplication \( AB \) is only valid when the number of columns in the first matrix \( A \) equals the number of rows in the second matrix \( B \).
In our example:

• Matrix \( A \) is \( 3 \times 1 \) and Matrix \( B \) is \( 1 \times 3 \). Here, multiplication \( AB \) results in a new \( 3 \times 3 \) matrix because the inner dimensions (1 and 1) match. \[AB = \begin{bmatrix} 7 \ 8 \ 9 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 7 & 14 & 21 \ 8 & 16 & 24 \ 9 & 18 & 27 \end{bmatrix}\]
• However, when attempting \( A^2 \), it is undefined because \( A \) is a column vector and doesn't have a second dimension equal to its own row count.

The key here is ensuring that the provided matrix dimensions enable valid multiplication.

Matrix addition and subtraction involve element-wise operations, meaning you add or subtract corresponding elements from two matrices. However, these operations require the matrices to have identical dimensions.

In the exercise, the operation \( A - 2B \) was attempted. First, \( 2B \) is calculated:

• \[2B = 2 \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} = \begin{bmatrix} 2 & 4 & 6 \end{bmatrix}\]

However, subtraction involving \( A \) and \( 2B \) is undefined because their dimensions differ - \( A \) is \( 3 \times 1 \) and \( 2B \) is \( 1 \times 3 \).
Always check matrix dimensions before attempting addition or subtraction to confirm the operation is valid.

Vectors are a type of matrix with either only one row or one column. In this exercise, we worked with one column vector (\( A \)) and one row vector (\( B \)). Vector operations are subject to matrix operation rules.

Since vectors can be considered as simplified matrices, we apply matrix rules similarly. Here:

• Multiplying a vector by a scalar changes each component of the vector proportionally, just as \( 3A \) did.
• Adding or subtracting vectors is only possible when both vectors have the same dimensions.

When multiplying a row vector by a column vector (or vice versa), as seen in \( BA \), the result is a single number or a \( 1 \times 1 \) matrix, often called the dot product:\[BA = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix} \begin{bmatrix} 7 \ 8 \ 9 \end{bmatrix} = \begin{bmatrix} 50 \end{bmatrix}\] This simple product involves matching dimensions as with any other matrix multiplication.