Problem 77 - College Algebra and Calculus: An Applied Approach

Question

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} 4 & 0 \\ 3 & -2 \end{array}\right], \quad B=\left[\begin{array}{ll} -1 & 1 \\ -2 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

The solution to the exercise is (a) -8, (b) 0, (c) \[\[\begin{array}{rr} -4 & 4 \ 0 & 2 \end{array}\]\], (d) -8

Step 1: Find the determinant of matrix A

We can find the determinant \(|A|\) of a 2x2 matrix \[A = \[\begin{array}{rr} a & b \ c & d \end{array}\]\] by using the formula \(|A| = ad - bc\). Substituting the values from matrix A, we get \(|A| = 4*(-2) - 0*3 = -8\)

Step 2: Find the determinant of matrix B

Using the same formula as step 1, we substitute the values from matrix B which gives us \(|B| = -1*2 - 1*(-2) = -2 + 2 = 0\)

Step 3: Multiply matrix A and matrix B

To multiply two matrices, we take each row from the first matrix and multiply and add up corresponding elements with each column of the second matrix. Doing this, we get \(AB = \[\begin{array}{rr} -4 & 4 \ 0 & 2 \end{array}\]\)

Step 4: Find the determinant of the resulting matrix AB

Using the formula from step 1, we find the determinant \(|AB| = -4*2 - 4*0 = -8\)

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Understanding this will make many matrix-related tasks easier. When you multiply two matrices, such as matrices \( A \) and \( B \), you need to follow a specific method to get each element in the resulting matrix.
Think of it like combining each row of the first matrix with each column of the second matrix. Here's how it works practically:

• Take the first row of matrix \( A \).
• Multiply each element of this row by the corresponding element in the first column of matrix \( B \).
• Add all those products. The result will be the element in the first row, first column of the new matrix \( AB \).
• Repeat this process using the same row of \( A \) with the next column of \( B \), and so on.

This process is repeated for each row of \( A \), moving through all the columns in \( B \) until the new matrix \( AB \) is completed. In the given exercise, when multiplying \( A \) and \( B \), you form the resulting matrix \( AB = \begin{bmatrix} -4 & 4 \ 0 & 2 \end{bmatrix} \). Each element of this matrix is derived from the multiplication and addition steps described above.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, and understanding it is essential to grasping more complex matrix concepts. It consists of two rows and two columns, containing four elements in total. An example form of a 2x2 matrix is \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \).
What makes 2x2 matrices particularly important is their simplicity, which makes them an excellent introduction to matrix operations such as addition, subtraction, and multiplication. Additionally, the rules for calculating determinants and inverses are easier to apply with a 2x2 matrix compared to larger matrices, making them a perfect starting point for beginners.
2x2 matrices are frequently used in various applications, from solving systems of equations to transformations in computer graphics. By working through the mentioned exercise, you can see how various operations such as finding determinants or performing multiplications are performed on 2x2 matrices, offering a foundation for understanding more complex matrices.

Determinant Formula

The determinant is a special number that can be calculated from a square matrix. For a 2x2 matrix like \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), the determinant is calculated using the formula \(|A| = ad - bc\).
This formula yields a single number that provides insights into the properties of the matrix. In essence, the determinant helps us understand whether a matrix is invertible (or non-invertible when the determinant is zero), and it directly relates to the area or volume scale change represented by the matrix.

• If the determinant is non-zero, the matrix has an inverse.
• If the determinant is zero, the matrix does not have an inverse and is singular.

In the context of the exercise provided, you compute the determinant of matrices \( A \) and \( B \), as well as the matrix product \( AB \). For matrix \( A \), \(|A| = -8\), indicating it's invertible. For \( B \), \(|B| = 0\), showing it's not invertible. Finding the determinant of the product also highlights that \(|AB| = -8\) suggests an invertible nature of the resulting matrix if individual factors allow it.