Problem 78 - College Algebra and Calculus: An Applied Approach

Question

Find \((a)|A|\), (b) \(|B|\), (c) \(A B\), and \((d)\) \(|A B|\). $$ A=\left[\begin{array}{rr} 5 & 4 \\ 3 & -1 \end{array}\right], \quad B=\left[\begin{array}{rr} 0 & 6 \\ 1 & -2 \end{array}\right] $$

Step-by-Step Solution

Answer

(a) The determinant of A, |A| is -17. (b) The determinant of B, |B| is -6. (c) The matrix multiplication AB gives [4, 22; -1,20]. (d) The determinant of AB, |AB| is 102.

Step 1: Calculate the determinant of A

The determinant of a 2x2 matrix \(\left[\begin{array}{cc}a & b \c & d\end{array}\right]\) is given by \(ad - bc\). Applying this to the coefficients of matrix A gives: |A|= (5*-1) - (4*3) = -5 - 12 = -17

Step 2: Calculate the determinant of B

Using the same formula for the determinant, |B|= (0*-2) - (6*1) = 0 - 6 = -6

Step 3: Compute the multiplication AB

Matrix multiplication is carried out as follows: The elements in the first row of the resulting matrix are obtained by multiplying the elements of the first row of the first matrix by the corresponding elements in the first column of the second matrix, and then adding them. This process is repeated for the second row, using the second column of the second matrix. Using this rule on our given matrices A and B, we get AB = [(5*0 + 4*1), (5*6 + 4*-2); (3*0 - 1*1), (3*6 - 1*-2)] = [4, 22; -1,20]

Step 4: Calculate the determinant of AB

Lastly, we calculate the determinant of the matrix resulting from the multiplication in the previous step using the determinant rule for a 2x2 matrix which leads to: |AB|= (4*20) - (22*-1) = 80 - -22 = 102

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental concept in linear algebra, and understanding it is crucial for working with matrices in various applications. Matrix multiplication involves combining two matrices to produce another matrix. The rule for multiplying two matrices involves the following key points:

• Only matrices where the number of columns in the first matrix matches the number of rows in the second can be multiplied together.
• The resulting matrix will have dimensions that match the number of rows of the first matrix and the number of columns of the second matrix.
• To find an element in the resulting matrix, multiply the elements of the corresponding row from the first matrix by the elements of the corresponding column from the second matrix, then sum these products.

This process allows for complex operations to be represented and calculated using matrices in fields such as computer graphics, systems theory, and various engineering disciplines.

2x2 Matrices

A 2x2 matrix is one of the simplest forms of a matrix, consisting of two rows and two columns. This matrix can be expressed in the form:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]2x2 matrices are often used to represent transformations in a two-dimensional space. They make calculations like matrix multiplication and finding determinants quite straightforward. Here's how you can understand them further:

• Determinant: The determinant of a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as \( ad - bc \). This value helps determine if a matrix is invertible (non-zero determinant) or singular (zero determinant).
• Matrix Properties: 2x2 matrices can represent various linear transformations including rotation, scaling, and shearing in two-dimensional space.
• Matrix Operations: Due to their size, operations like addition, subtraction, and multiplication are easier and computationally simpler compared to larger matrices.

Understanding the structure and operations of 2x2 matrices is very helpful as it provides a foundation for working with larger matrices and matrix operations.

Linear Algebra

Linear algebra is a branch of mathematics focusing on vectors, vector spaces, and linear transformations represented via matrices. It serves as a critical area of mathematics with applications in various scientific and engineering fields. Here's a closer look at some essential aspects:

• Vectors and Scalar Multiplication: At the heart of linear algebra are vectors and scalar multiplication. Vectors are like arrows, having both magnitude and direction, while scalar multiplication involves stretching or shrinking vectors.
• Matrix Representation: Matrices are used to represent systems of linear equations, making them very useful for solving various types of algebraic problems. They help in transformations and changing coordinate systems.
• Determinants and Inverses: Determinants play a pivotal role in matrix operations, especially when finding the inverse of a matrix. An invertible matrix (non-zero determinant) allows for solutions to linear systems to be computed easily.
• Applications: Linear algebra is applied in computer graphics for rendering images, machine learning for optimization, and engineering for calculating forces in physical systems.

Grasping linear algebra concepts enhances the understanding of many practical problems, providing tools for solving complex systems in numerous disciplines.