Problem 65 - Precalculus with Limits

Question

In Exercises 63-70, find (a) \(|A|\), (b) \(|B|\), (c) \(AB\), and (d) \(|AB|\). \(A = \left[ \begin{array}{r} 4 & 0 \\ 3 & -2 \end{array} \right]\), \(B = \left[ \begin{array}{r} -1 && 1 \\ -2 && 2 \end{array} \right]\)

Step-by-Step Solution

Answer

(a) \(|A| = -8\), (b) \(|B| = -2\), (c) \(AB = \left[ \begin{array}{cc} -4 & 4 \\ 1 & -1 \end{array} \right]\), (d) \(|AB| = 0\)

Step 1: Compute the determinant of matrix A

First, calculate the determinant of matrix A. The determinant of a 2x2 matrix \( \left[ \begin{array}{cc} a & b \\ c & d \end{array} \right] \) is given by \(ad - bc\). So, \(|A| = (4*-2) - (0*3) = -8.

Step 2: Compute the determinant of matrix B

Next, calculate the determinant of matrix B. So, \(|B| = (-1*2) - (1*-2) = -2.

Step 3: Multiply the matrices A and B

Now, let's multiply the matrices A and B. For two matrices to be multiplied, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In order to calculate \(AB\), each element of the first row of the first matrix should be multiplied with the corresponding element in the first column of the second matrix, and the results should be summed up to get the first element of the resulting matrix. This operation should be repeated for all elements of all the rows of the first matrix with all the columns of the second matrix. Therefore, \(AB = \left[ \begin{array}{cc} 4*-1 + 0*-2 & 4*1 + 0*2 \\ 3*-1 + -2*-2 & 3*1 + -2*2 \end{array} \right] = \left[ \begin{array}{cc} -4 & 4 \\ 1 & -1 \end{array} \right]\).

Step 4: Compute the determinant of the result matrix AB

Finally, let's find the determinant of the result matrix \(AB\). So, \(|AB| = (-4*-1) - (4*1) = 0.

Key Concepts

Determinant

The determinant is a special number that can be calculated from a square matrix. It provides important information about the matrix, such as whether it is invertible. Let's dive into how this works for a 2x2 matrix like matrix A.

For a 2x2 matrix \(\left[ \begin{array}{cc} a & b \ c & d \end{array} \right]\), the determinant is found using the formula \(ad - bc\). This gives us an easy way to compute the determinant without extra steps:

• Multiply the leading diagonal elements \(a\) and \(d\).
• Multiply the other diagonal elements \(b\) and \(c\).
• Subtract the second product from the first, resulting in \(ad - bc\).

In our exercise, the determinant of matrix A is \((4 \times (-2)) - (0 \times 3) = -8\). Similarly, for matrix B, it is \((-1 \times 2) - (1 \times (-2)) = -2\). These determinants help us understand the properties of matrices A and B.

Matrix Multiplication

Matrix multiplication is crucial when working with linear algebra as it allows us to combine information from different matrices. The operation, while simple, follows strict rules.

Here’s how you perform matrix multiplication for matrices A and B:

• The number of columns in the first matrix should match the number of rows in the second.
• For each element in the resulting matrix, multiply elements from the rows of the first matrix with the respective elements of the columns in the second matrix.
• Sum the products for each pair, positioning this sum at the corresponding spot in the product matrix.

With our matrices, we calculate \(AB\) as follows: by first multiplying and then summing appropriately, we obtain the product matrix, \(\left[ \begin{array}{cc} -4 & 4 \ 1 & -1 \end{array} \right]\). This process concatenates the information held within both matrices into one matrix.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. Despite its simplicity, it is a powerful tool in linear algebra.

In a 2x2 matrix:

• There are four individual elements, typically labeled as \( \left[ \begin{array}{cc} a & b \ c & d \end{array} \right] \).
• It can represent transformations in two-dimensional space, such as rotations, scaling, and shearing.
• The determinant of this matrix is a quick yardstick to determine invertibility and the type of transformation it represents.

For instance, matrices A and B in our exercise are simple 2x2 matrices. Despite their simplicity, working through the calculation of these matrices helps build a foundation for tackling more complicated matrix operations.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It's a fundamental discipline that underpins various applications across sciences and engineering.

Understanding linear algebra involves several key topics:

• Matrices and Determinants: These allow us to represent and solve systems of linear equations efficiently.
• Matrix Multiplication: Essential for modeling and transforming complex systems, such as in computer graphics.
• Vector Spaces: Concepts that expand beyond numbers to describe positions, transformations, and functions.

Matrix operations like those demonstrated in this exercise facilitate the analysis and understanding of vector transformations. As such, grasping the mechanics of matrix determinants and multiplication builds a solid basis for further study in linear algebra and its applications.