Problem 45 - College Algebra with Modeling and Visualization

Question

Choose two matrices \(A\) and \(B\) with dimension \(2 \times 2\). Calculate det \(A\), det \(B\), and \(\operatorname{det}(A B)\). Repeat this process until you are able to discover how these three determinants are related. Summarize your results.

Step-by-Step Solution

Answer

\(\det(AB) = \det(A) \times \det(B)\).

Step 1: Choose matrices A and B

Let's choose two matrices \(A\) and \(B\) with dimensions \(2 \times 2\). \(A = \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 \ 5 & 2 \end{bmatrix}\). Now, we'll find the determinants of these matrices.

Step 2: Calculate det A

The determinant of a \(2 \times 2\) matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) is calculated as \( \det(A) = ad - bc \). For matrix \(A = \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix}\), \[ \det(A) = (2 \times 4) - (3 \times 1) = 8 - 3 = 5. \]

Step 3: Calculate det B

Using the same formula, calculate the determinant of matrix \(B = \begin{bmatrix} 1 & 0 \ 5 & 2 \end{bmatrix}\), \[ \det(B) = (1 \times 2) - (0 \times 5) = 2 - 0 = 2. \]

Step 4: Calculate AB

Multiply matrices \(A\) and \(B\) to find their product \(AB\).\[ AB = \begin{bmatrix} 2 & 3 \ 1 & 4 \end{bmatrix} \begin{bmatrix} 1 & 0 \ 5 & 2 \end{bmatrix} = \begin{bmatrix} 2 \times 1 + 3 \times 5 & 2 \times 0 + 3 \times 2 \ 1 \times 1 + 4 \times 5 & 1 \times 0 + 4 \times 2 \end{bmatrix} \]\[ = \begin{bmatrix} 17 & 6 \ 21 & 8 \end{bmatrix}. \]

Step 5: Calculate det(AB)

Now find the determinant of the matrix \(AB = \begin{bmatrix} 17 & 6 \ 21 & 8 \end{bmatrix}\), \[ \det(AB) = (17 \times 8) - (6 \times 21) = 136 - 126 = 10. \]

Step 6: Compare and Analyze

Let's compare \( \det(A) \times \det(B) \) with \( \det(AB) \). We have:\[ \det(A) \times \det(B) = 5 \times 2 = 10 \]\[ \det(AB) = 10 \]These results show us that \( \det(AB) = \det(A) \times \det(B) \).

Step 7: Generalize the Findings

Based on the calculation for the chosen matrices, the determinant of the product of two matrices \(A\) and \(B\) is equal to the product of their determinants. This can be summarized as \( \det(AB) = \det(A) \times \det(B) \).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that involves combining two matrices to produce a third matrix. To understand how this works, we need to follow a specific process. Let's consider matrix multiplication using two matrices, each with dimensions \(2 \times 2\). For such matrices, if we have \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and \(B = \begin{bmatrix} e & f \ g & h \end{bmatrix}\), the resulting matrix \(AB\) is computed as follows:

• The element in the first row, first column is \(ae + bg\).
• The element in the first row, second column is \(af + bh\).
• The element in the second row, first column is \(ce + dg\).
• The element in the second row, second column is \(cf + dh\).

This operation is not commutative, meaning that \(AB\) is generally not equal to \(BA\). Understanding this is crucial as it influences how determinants behave during multiplication.

2x2 Matrices

A \(2x2\) matrix is a simple yet powerful tool in linear algebra with two rows and two columns. This type of matrix is written in the form \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), where \(a\), \(b\), \(c\), and \(d\) are its elements. The simplicity of \(2x2\) matrices makes them ideal for introductory studies in matrix operations, such as addition, multiplication, and finding determinants.

Calculating a determinant for a \(2x2\) matrix is straightforward: it's given by the formula \(ad - bc\). Despite their simplicity, \(2x2\) matrices can represent a range of transformations such as scaling and rotation in a plane, making them practical for various applications.

Determinant Properties

The determinant is a special number that can be calculated from a square matrix. It has properties that reveal a lot about the matrix itself, particularly concerning invertibility and scaling transformations. For a \(2x2\) matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\), the determinant is \(ad - bc\).

Let's delve into some important properties of determinants:

• If the determinant of a matrix is zero, the matrix does not have an inverse and is said to be singular.
• The determinant of a product of two matrices equals the product of their determinants: \(\det(AB) = \det(A) \times \det(B)\). This is a valuable property particularly in simplifying the calculation of determinants for large matrices.
• Interchanging any two rows (or columns) of a matrix results in a determinant with the opposite sign.

These properties aid in understanding and solving linear equations, transforming geometric entities, and more.