Problem 21 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{r}{6} \\ {-3}\end{array}\right] \cdot\left[\begin{array}{ll}{2} & {-7}\end{array}\right]\)

Step-by-Step Solution

Answer

The product is a 2x2 matrix: \(\begin{bmatrix} 12 & -42 \\ -6 & 21 \end{bmatrix}\).

Step 1: Understanding the Matrices

We are given two matrices: 1. A column matrix \[\begin{bmatrix}6 \-3\end{bmatrix}\]2. A row matrix \[\begin{bmatrix}2 & -7\end{bmatrix}\]We need to find their product.

Step 2: Check Compatibility for Multiplication

For matrix multiplication, if matrix A is of size \(m \times n\) (meaning it has m rows and n columns) and matrix B is of size \(n \times p\), then the resulting matrix will be of size \(m \times p\). - Our column matrix is \(2 \times 1\) and the row matrix is \(1 \times 2\), so they can be multiplied. The resulting matrix will be of size \(2 \times 2\).

Step 3: Perform Matrix Multiplication

To multiply a column matrix by a row matrix, produce a matrix where each entry is the product of the corresponding entry from the column and the row:\[\begin{bmatrix}6 \-3\end{bmatrix}\times\begin{bmatrix}2 & -7\end{bmatrix}= \begin{bmatrix}6 \times 2 & 6 \times (-7)\-3 \times 2 & -3 \times (-7)\end{bmatrix}= \begin{bmatrix}12 & -42 \-6 & 21\end{bmatrix}\]

Step 4: Interpret the Result

The resulting matrix from the multiplication is:\[\begin{bmatrix}12 & -42 \-6 & 21\end{bmatrix}\]. This is a 2x2 matrix formed by multiplying each element of the given column matrix with each element of the row matrix.

Key Concepts

Column Matrix

A column matrix is a type of matrix that consists of a single column. This means that it has multiple rows, but only one column of numbers. It's like a vertical list of numbers. For instance, the column matrix given in our example is:

• \(\begin{bmatrix} 6 \ -3 \end{bmatrix}\)

This column matrix has two entries arranged in one column, making it a \( 2 \times 1 \) matrix. The first number, 6, is in the first row, and the second number, -3, is right below it in the second row.
Column matrices are fundamental in matrix multiplication because they are easy to multiply with row matrices, allowing us to form new matrices through their interactions.

Row Matrix

A row matrix is essentially the opposite of a column matrix. Instead of being a single column of numbers, it's a single row of numbers. The row matrix in our exercise looks like this:

• \(\begin{bmatrix} 2 & -7 \end{bmatrix}\)

This row matrix is a \( 1 \times 2 \) matrix. The numbers are arranged horizontally in one row and two columns. The sequence is straightforward, with 2 in the first column followed by -7 in the second column.
Row matrices are crucial in linear algebra because, when multiplied by column matrices, they help form all the entries in the resulting matrix. This process is a simple yet powerful way to manipulate and transform data in matrix form.

Matrix Dimensions

Matrix dimensions refer to the way the matrix is sized in terms of rows and columns. Notated as \( m \times n \), "m" represents the number of rows and "n" represents the number of columns. For effective matrix multiplication, understanding dimensions is key.
In our example:

• The column matrix has dimensions \( 2 \times 1 \) because it has 2 rows and 1 column.
• The row matrix has dimensions \( 1 \times 2 \) because it has 1 row and 2 columns.

When we need to calculate a matrix product, the inner dimensions must match, meaning the number of columns in the first matrix must equal the number of rows in the second. Here, the inner dimension of both matrices is 1, making the multiplication possible.

Multiplication Compatibility

Matrix multiplication is not always possible. The first step in trying to multiply two matrices is to confirm that they are compatible for multiplication. Compatibility depends on matrix dimensions.
For any two matrices, say A and B, where A is an \( m \times n \) matrix and B is an \( n \times p \) matrix, they can be multiplied if and only if the number of columns in the first matrix (A) matches the number of rows in the second matrix (B).

• In our example, a \( 2 \times 1 \) column matrix is multiplied by a \( 1 \times 2 \) row matrix.
• The common dimension is 1 (the column count of the first = row count of the second), allowing multiplication.
• The result is a \( 2 \times 2 \) matrix, because the number of rows from the column matrix (2) and the number of columns from the row matrix (2) determine the size of the resulting matrix.

Always ensure matrices are multiplication compatible before proceeding; otherwise, the product is undefined.