Problem 59 - Algebra 2

Question

Find each product, if possible. $$ \left[\begin{array}{lll}{4} & {2} & {0}\end{array}\right] \cdot\left[\begin{array}{rr}{3} & {-2} \\ {1} & {0} \\ {5} & {6}\end{array}\right] $$

Step-by-Step Solution

Answer

The product is \(\begin{bmatrix} 14 & -8 \end{bmatrix}\).

Step 1: Understanding Matrix Multiplication

In order to multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, we have a 1x3 matrix (1 row, 3 columns) and a 3x2 matrix (3 rows, 2 columns). Since 3 columns of the first matrix equal 3 rows of the second, the multiplication is possible.

Step 2: Performing the Matrix Multiplication

To multiply the matrices, take the row vector of the first matrix, and multiply each element with corresponding elements in each column of the second matrix, summing up those products. Let the first matrix be \([4, 2, 0]\) and the second matrix be \[\begin{bmatrix} 3 & -2 \ 1 & 0 \ 5 & 6 \end{bmatrix}\].

Step 3: Calculating the First Element

Calculate the first element of the resulting matrix by multiplying the first row with the first column: \(4 \times 3 + 2 \times 1 + 0 \times 5 = 12 + 2 + 0 = 14\).

Step 4: Calculating the Second Element

Calculate the second element of the resulting matrix by multiplying the first row with the second column: \(4 \times (-2) + 2 \times 0 + 0 \times 6 = -8 + 0 + 0 = -8\).

Step 5: Formulating the Resultant Matrix

Combine the results from the calculations to form the resulting matrix: \[\begin{bmatrix} 14 & -8 \end{bmatrix}\].

Key Concepts

Matrices

A matrix is an organized collection of numbers arranged in rows and columns. It is usually enclosed in brackets, and each number in the matrix is referred to as an element. Matrices can vary in size and shape, which is determined by their number of rows and columns. For instance, a matrix with three rows and two columns is called a 3x2 matrix. Each matrix serves as a foundational structure for performing various mathematical operations, such as addition, subtraction, and multiplication. Matrices are essential in various fields including physics, engineering, computer science, and statistics. They also help in solving complex linear equations, making them powerful tools in advanced mathematics.

Matrix Dimensions

Matrix dimensions are a critical aspect of understanding and working with matrices. The dimension of a matrix is denoted by the number of rows (m) times the number of columns (n), written as \( m\times n \). This size notation helps in identifying the structure and organization of a matrix.

• An example is the matrix \( \left[\begin{array}{rr}{3} & {-2} \ {1} & {0} \ {5} & {6}\end{array}\right] \), which has dimensions 3x2.
• It contains 3 rows and 2 columns.

When multiplying matrices, the matrix dimensions must satisfy certain conditions. Specifically, the number of columns of the first matrix must equal the number of rows of the second matrix. This ensures that each element of the row vector can be paired with the corresponding element in the column.

Matrix Product Calculation

Matrix product calculation involves multiplying rows of one matrix by the columns of another. If the dimensions align for multiplication, the resultant matrix dimensions will be determined by the number of rows in the first matrix and the number of columns in the second matrix. For example, if we have a 1x3 matrix \([4, 2, 0]\) and a 3x2 matrix \( \left[\begin{array}{rr}{3} & {-2} \ {1} & {0} \ {5} & {6}\end{array}\right] \).The resulting matrix will have dimensions 1x2. The calculation process involves:

• Multiplying each element of the row vector \([4, 2, 0]\) by corresponding elements in the columns of the second matrix.
• Summing these products for each column.

The matrix product in this case results in \( \left[\begin{array}{rr}{14} & {-8}\end{array}\right] \), where each entry is calculated by the dot product of the row from the first matrix and a column from the second matrix. Such calculations are fundamental in transforming data and solving system equations effectively.