Problem 22 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{rr}{3} & {-2} \\ {5} & {1}\end{array}\right] \cdot\left[\begin{array}{ll}{4} & {1} \\ {2} & {7}\end{array}\right]\)

Step-by-Step Solution

Answer

The product matrix is \[ \left[\begin{array}{cc} 8 & -11 \\ 22 & 12 \end{array}\right] \].

Step 1: Understand Matrix Multiplication Criteria

Matrix multiplication is only possible when the number of columns in the first matrix matches the number of rows in the second matrix. Here, both matrices are 2x2, so multiplication is possible.

Step 2: Setup Matrix Multiplication

For matrix multiplication, use the formula: \[C_{ij} = \sum_{k=1}^{n} A_{ik}B_{kj}\]where \(C_{ij}\) is the element of the resulting matrix, \(A_{ik}\) is an element from row \(i\) of the first matrix \(A\), and \(B_{kj}\) is an element from column \(j\) of the second matrix \(B\).

Step 3: Calculate Element C11

For \(C_{11}\): Multiply the first row of the first matrix by the first column of the second matrix:\[C_{11} = 3(4) + (-2)(2) = 12 - 4 = 8\]

Step 4: Calculate Element C12

For \(C_{12}\): Multiply the first row of the first matrix by the second column of the second matrix:\[C_{12} = 3(1) + (-2)(7) = 3 - 14 = -11\]

Step 5: Calculate Element C21

For \(C_{21}\): Multiply the second row of the first matrix by the first column of the second matrix:\[C_{21} = 5(4) + 1(2) = 20 + 2 = 22\]

Step 6: Calculate Element C22

For \(C_{22}\): Multiply the second row of the first matrix by the second column of the second matrix:\[C_{22} = 5(1) + 1(7) = 5 + 7 = 12\]

Step 7: Combine Results into Product Matrix

Place all computed elements into the resulting matrix: \[\left[\begin{array}{cc} 8 & -11 \ 22 & 12 \end{array}\right]\]

Key Concepts

2x2 matrices

A 2x2 matrix is a rectangular array of numbers with 2 rows and 2 columns. These matrices are often used in mathematics to perform various operations, such as matrix addition, subtraction, and multiplication. The general structure of a 2x2 matrix is given by:
\[\left[\begin{array}{cc}a_{11} & a_{12} \a_{21} & a_{22}\end{array}\right]\]where \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\) are the individual elements of the matrix. Each element can be any real or complex number, and the matrix as a whole is an organized way of representing data or equations in a compact form. Understanding the layout of a 2x2 matrix is fundamental to utilizing matrices in problems involving transformations, vector spaces, and systems of equations.

elements of matrix

The elements of a matrix are the individual values or numbers that fill the matrix's rows and columns. In a 2x2 matrix, there are four elements, typically denoted as \(a_{11}\), \(a_{12}\), \(a_{21}\), and \(a_{22}\).

• \(a_{11}\) is the element in the first row and first column.
• \(a_{12}\) is the element in the first row and second column.
• \(a_{21}\) is the element in the second row and first column.
• \(a_{22}\) is the element in the second row and second column.

Each element plays a crucial role when performing operations such as matrix multiplication, where specific elements are used to compute components of the resulting matrix. The organization of these elements allows for systematic calculations, following a set pattern.

resulting matrix

The resulting matrix from a multiplication operation between two matrices is derived by following a defined mathematical procedure. Specifically, each element in the resulting 2x2 matrix is computed from the elements of the two original matrices through a process of multiplication and addition.
For this exercise:

• The element in the first row, first column \(C_{11}\), is calculated by multiplying the corresponding elements of the first row of Matrix A and the first column of Matrix B.
• Similarly, \(C_{12}\) is obtained using the first row of Matrix A and the second column of Matrix B.
• The element \(C_{21}\) utilizes the second row of Matrix A and the first column of Matrix B.
• Finally, \(C_{22}\) is calculated from the second row of Matrix A and the second column of Matrix B.

The organization of calculations helps ensure accuracy and consistency, leading to a complete and precise resulting matrix. This matrix reflects how inputs combine within a system's specified constraints.

matrix multiplication criteria

Matrix multiplication has specific criteria that must be met for the operation to be performed correctly. Primarily, for two matrices to be multiplied, the number of columns in the first matrix must match the number of rows in the second matrix, forming a valid pairing.
For example, when multiplying two 2x2 matrices, the structure of the problem fits well within these criteria, as both matrices have compatible dimensions:

• Both matrices have 2 columns and 2 rows, satisfying the multiplication condition.
• This alignment allows each row from the first matrix to be multiplied by each column from the second matrix, leading to a 2x2 resulting matrix.

Understanding and ensuring that your matrices meet the multiplication criteria is critical before starting any calculations. This prevents errors and guarantees that the multiplication process can be completed successfully, providing a usable result in various applications from mathematics to engineering.