Problem 14 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{r}{4} \\ {1} \\\ {-3}\end{array}\right]+\left[\begin{array}{r}{6} \\ {-5} \\\ {8}\end{array}\right] $$

Step-by-Step Solution

Answer

The result of the matrix addition is \( \begin{bmatrix} 10 \\ -4 \\ 5 \end{bmatrix} \).

Step 1: Identify the Matrices

The first matrix is \( \begin{bmatrix} 4 \ 1 \ -3 \end{bmatrix} \) and the second matrix is \( \begin{bmatrix} 6 \ -5 \ 8 \end{bmatrix} \). Both matrices are column vectors with the same dimensions, making addition possible.

Step 2: Add Corresponding Elements

Add the corresponding elements of the two matrices: \[ 4 + 6,\ 1 + (-5),\ (-3) + 8 \].

Step 3: Simplify Each Sum

Calculate the sums for each pair of elements: \[ 4+6=10,\ 1+(-5)=-4,\ (-3)+8=5 \].

Step 4: Write the Resultant Matrix

Combine the simplified sums into a new matrix: \( \begin{bmatrix} 10 \ -4 \ 5 \end{bmatrix} \).

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation in linear algebra that involves adding two matrices by summing their corresponding elements. This operation requires both matrices to have the same dimensions, which means they must have the same number of rows and columns. When matrices meet this criterion, matrix addition is straightforward and enables many applications, such as solving linear systems or combining transformations.

Matrix addition works in a step-by-step manner:

• Identify matrices and their dimensions.
• Ensure that matrices are of the same size.
• Add corresponding elements to form a new matrix.
• Each entry in the resulting matrix is the sum of the entries at the same position in the original matrices.

For example, if we have two matrices, each being a column vector of size 3x1, such as \[ egin{bmatrix} 4 \ 1 \ -3 \ ext{and} \ egin{bmatrix} 6 \ -5 \ 8 \ ight] \ ight] \ , \\]we can add them by summing each pair of corresponding elements, leading to another column vector: \[egin{bmatrix} 10 \ -4 \ 5 \ ight] \\].

Column Vector

A column vector is a type of matrix that consists of a single column of elements, typically arranged vertically. Each element of the column vector represents an entry in a sequence or series of data points. Column vectors are widely used in mathematics, particularly in vector algebra and matrix operations.

Key points about column vectors:

• They are essentially matrices with a single column and one or more rows.
• They are often used to represent data, coordinates, and variables in equations.
• Operations like matrix addition, scalar multiplication, and vector dot product can be performed with them.
• In terms of dimensions, a column vector can be represented as "n x 1", where "n" is the number of elements.

For instance, the column vector \[ egin{bmatrix} 4 \ 1 \ -3 \ ight] \] is a 3x1 vector, meaning it has 3 elements arranged in a single column.

Matrix Dimensions

Matrix dimensions refer to the number of rows and columns that a matrix contains, usually expressed in the form "m x n", where "m" is the number of rows and "n" is the number of columns. Understanding matrix dimensions is crucial for performing various matrix operations, including addition, multiplication, and matrix inversion.

Key aspects of matrix dimensions include:

• Two matrices can only be added if they have the same dimensions.
• For matrix multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.
• The dimension helps in classifying the matrix as a square matrix (m = n), row vector (1 x n), or column vector (n x 1).

In the context of the given problem, both matrices involved in the addition have dimensions 3x1, characteristic of column vectors. Thus, the addition is feasible, and we can straightforwardly perform the operation as each dimension matches between the matrices involved.