Problem 12 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The product of the matrices is \[ \left[\begin{array}{rr}{1} & {0} \ {2} & {-3}\end{array}\right]\left[\begin{array}{rr}{-3} & {4} \ {5} & {2}\end{array}\right] = \left[\begin{array}{rr}{-3} & {4} \ {-21} & {-2}\end{array}\right]\]

Step 1: Understand Matrix Multiplication

To multiply two matrices, calculate each element of the product matrix by taking the dot product (element-by-element product and sum) of the corresponding row of the first matrix with the corresponding column of the second matrix.

Step 2: Multiply the First Row of Matrix A by Each Column of Matrix B

For the element at the first row and first column of the product matrix, multiply each element of the first row of matrix A by the corresponding element of the first column of matrix B, then sum up the products. Repeat this process for the first row and second column of the product matrix.

Step 3: Multiply the Second Row of Matrix A by Each Column of Matrix B

For the element at the second row and first column of the product matrix, multiply each element of the second row of matrix A by the corresponding element of the first column of matrix B, then sum up the products. Repeat this process for the second row and second column of the product matrix.

Step 4: Write Down the Resulting Matrix

Combine all the elements obtained from the previous steps into a new matrix. This is the product of the two given matrices.

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a fundamental operation used in matrix multiplication. It involves multiplying corresponding elements of two sequences of numbers and then summing those products to yield a single number. For matrices, the dot product is taken between the rows of one matrix and the columns of another.

For example, if we have two vectors, \( \vec{a} = (a_1, a_2) \) and \( \vec{b} = (b_1, b_2) \), the dot product is calculated as \( \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 \).

When applying this to matrix multiplication, each element of the resulting matrix is a dot product between a row of the first matrix and a column of the second matrix. It's essential to ensure that the number of columns in the first matrix matches the number of rows in the second matrix to perform this operation.

Matrix Product

Matrix product, or matrix multiplication, involves combining two algebraic matrices to produce a third matrix. Unlike element-wise multiplication, the matrix product is more complex and follows a specific rule set.

The general rule for matrix multiplication is that if we want to multiply an \( m \times n \) matrix (let's call it Matrix A) by an \( n \times p \) matrix (Matrix B), the result will be an \( m \times p \) matrix. To obtain each element of the product, we take the dot product of the corresponding row from Matrix A and column from Matrix B.

This operation is significant in many areas of mathematics, physics, engineering, and computer science, underscoring relationships and transformations represented by matrices.

Elementary Row Operations

Elementary row operations are tools for manipulating matrices to simplify complex problems, especially when solving systems of linear equations or finding the inverse of a matrix. There are three types of elementary row operations:

• Row swapping: exchanging two rows within a matrix,
• Row multiplication: multiplying all elements of a row by a non-zero constant,
• Row addition: adding or subtracting the elements of two rows and replacing one of the rows with the result.

Understanding these operations is vital for various procedures in linear algebra, such as finding the reduced row echelon form of a matrix or the determinant.

Algebraic Matrices

Algebraic matrices are grids of numbers or expressions arranged in rows and columns that represent both numerical and abstract concepts in linear algebra. Each entry in a matrix can represent a coefficient of a linear equation, transformations in space, probabilities in stochastic processes, and more.

Matrices can undergo operations such as addition, subtraction, and multiplication, provided certain conditions are met. The power of matrices lies in their ability to succinctly represent complex systems and facilitate operations on those systems that would be cumbersome to handle individually. Mastery of algebraic matrix operations is essential for advancing in fields that require the modeling of complex, interconnected data.