Problem 23 - Algebra 2

Question

Find each product, if possible. \(\left[\begin{array}{rr}{-1} & {0} \\ {5} & {2}\end{array}\right] \cdot\left[\begin{array}{cc}{6} & {-3} \\ {7} & {-2}\end{array}\right]\)

Step-by-Step Solution

Answer

The product is \(\begin{bmatrix} -6 & 3 \\ 44 & -19 \end{bmatrix}\).

Step 1: Understand Matrix Multiplication

To multiply two matrices, we need to ensure 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. The resulting matrix will also be 2x2.

Step 2: Multiply Row by Column

To find each element in the product matrix, we take the dot product of rows from the first matrix with columns from the second matrix. The element in the first row, first column of the product matrix is calculated by: \[(-1)\cdot6 + 0\cdot7 = -6 + 0 = -6\]

Step 3: Find Other Elements in the Resulting Matrix

Continuing from Step 2, we calculate the other elements:- First row, second column: \[(-1)\cdot(-3) + 0\cdot(-2) = 3 + 0 = 3\]- Second row, first column:\[5\cdot6 + 2\cdot7 = 30 + 14 = 44\]- Second row, second column: \[5\cdot(-3) + 2\cdot(-2) = -15 - 4 = -19\]Thus, the resulting product matrix is:\[\left[\begin{array}{cc}-6 & 3 \ 44 & -19\end{array}\right]\]

Key Concepts

Dot Product

The dot product is a foundational concept in linear algebra, especially when dealing with matrices and vectors. It involves multiplying corresponding elements of two sequences of numbers and then summing these products.
For two given vectors, let's say \( \textbf{a} = [a_1, a_2]\) and \( \textbf{b} = [b_1, b_2]\), the dot product is calculated as:

• \(a_1 \cdot b_1 + a_2 \cdot b_2\)

In the context of matrices, the dot product is used to compute the elements of the product matrix. Here, we take a row vector from the first matrix and a column vector from the second matrix. Multiply corresponding numbers, sum them up, and place the result in the position within the resulting matrix. Understanding this concept is critical for performing matrix multiplication effectively.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. Each element of the matrix is identified by its position, usually written as \(a_{ij}\) where \(i\) is the row number and \(j\) is the column number.

• The general form of a 2x2 matrix is \(\begin{bmatrix}a_{11} & a_{12} \a_{21} & a_{22} \\end{bmatrix}\)

In problems involving 2x2 matrices, they are particularly simple to work with due to their small size, making them an excellent introduction into the world of matrix operations. When multiplying two 2x2 matrices, we derive four new elements, each calculated using the dot product of the relevant row and column.

Matrix Product

Matrix product, or matrix multiplication, is a process of combining two matrices to form a new matrix. This operation is distinct from the element-wise multiplication of matrices.
The matrix product \(C = AB\) is defined if the number of columns in \(A\) matches the number of rows in \(B\). The elements of \(C\) are calculated by taking the dot product of the appropriate row of \(A\) and column of \(B\).

• First matrix (A): dictates the number of rows in the product.
• Second matrix (B): dictates the number of columns in the product.

The practical importance of matrix product lies in its ability to represent complex transformations, systems of linear equations, and perform data operations efficiently in various fields like computer graphics, physics, and engineering.

Matrix Operations

Matrix operations encompass a variety of mathematical procedures that can be performed on matrices. These include addition, subtraction, scalar multiplication, and, most importantly, matrix multiplication (or matrix product).
Here's a brief look at each:

• Addition/Subtraction: For two matrices of the same dimensions, sum or subtract their corresponding elements.
• Scalar Multiplication: Multiply each element of the matrix by a constant (scalar).
• Multiplication: Only possible when the number of columns in the first matrix equals the number of rows in the second.

Understanding these operations is fundamental for manipulating data in linear algebra. They form the backbone of more complex operations and applications, whether you're solving equations or performing transformations in space.