Problem 20 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{ll}{2} & {-1}\end{array}\right] \cdot\left[\begin{array}{l}{5} \\ {4}\end{array}\right]\)

Step-by-Step Solution

Answer

The product of the matrices is 6.

Step 1: Understanding Matrix Dimensions

The first matrix is a 1x2 matrix: \( \begin{bmatrix} 2 & -1 \end{bmatrix} \), and the second matrix is a 2x1 matrix: \( \begin{bmatrix} 5 \4 \end{bmatrix} \). The inner dimensions are the same (2), which means matrix multiplication is possible.

Step 2: Setting Up Matrix Multiplication

To multiply the matrices, align the row from the first matrix with the column from the second matrix. This results in a single value from the operation: \( 2 \times 5 + (-1) \times 4 \).

Step 3: Performing the Calculations

Calculate each part of the expression: \( 2 \times 5 = 10 \) and \( (-1) \times 4 = -4 \). Sum these results to find the product of the matrices.

Step 4: Summing the Results

Add the results from the previous step: \( 10 + (-4) = 6 \).

Step 5: Finalizing the Answer

The result of multiplying the given matrices is a single value: \( 6 \).

Key Concepts

Matrix Dimensions

To understand matrix multiplication, first, we need to grasp the concept of matrix dimensions. The dimensions of a matrix are determined by the number of rows and columns it contains. Each matrix is written as an "m x n" matrix, where "m" is the number of rows and "n" is the number of columns.
For example, if a matrix has 3 rows and 2 columns, it is called a 3x2 matrix. In the exercise given, the first matrix is a 1x2 matrix: \( \begin{bmatrix} 2 & -1 \end{bmatrix} \), meaning it has 1 row and 2 columns. The second matrix is a 2x1 matrix: \( \begin{bmatrix} 5 \ 4 \end{bmatrix} \), which has 2 rows and 1 column.

• Matrix dimensions are crucial for determining if two matrices can be multiplied.
• The inner dimensions (the columns of the first matrix and the rows of the second matrix) must match for multiplication to be possible.

Matrix Operations

Matrix operations involve performing calculations between matrices to achieve a desired result. They include addition, subtraction, and multiplication.
Addition and subtraction are straightforward: they require matrices to be the same dimensions. Only identical matrices (having the same number of rows and columns) can be added or subtracted.

• Matrix addition: Sum up the corresponding elements in each matrix.
• Matrix subtraction: Subtract the corresponding elements.

Matrix multiplication is more complex. Unlike addition or subtraction, the dimensions of the matrices dictate whether they can be multiplied. You multiply along rows and columns to form the resulting matrix.
It’s important to align matrix dimensions properly to perform various matrix operations successfully.

Multiplication of Matrices

Multiplication of matrices involves more than just multiplying corresponding elements, as done in addition or subtraction. It's about pairing rows from the first matrix with columns from the second matrix. Here's how it works:

• Check Dimensions: Make sure the number of columns in the first matrix matches the number of rows in the second matrix. This is key for multiplication.
• Calculate Each Pair: Pair each row with each column to produce a new matrix's element. For each pair, multiply the corresponding elements and then sum up the products.

For the exercise, multiplying the given matrices with dimensions 1x2 and 2x1, you line up the row \([2, -1]\) with the column \(\begin{bmatrix} 5 \ 4 \end{bmatrix} \). The operations become: - \(2 \times 5 = 10\)- \((-1) \times 4 = -4\)Sum these results: \(10 + (-4) = 6\). Thus, the product is the scalar 6, resulting from this specific matrix multiplication.