Problem 38 - Precalculus: Mathematical for Calculus

Question

The matrices \(A, B, C, D, E, F,\) and \(G\) are defined as $$\begin{array}{l} A=\left[\begin{array}{rr} 2 & -5 \\ 0 & 7 \end{array}\right] \quad B=\left[\begin{array}{rrrr} 3 & \frac{1}{2} & 5 \\ 1 & -1 & 3 \end{array}\right] \quad C=\left[\begin{array}{rrr} 2 & -\frac{5}{2} & 0 \\ 0 & 2 & -3 \end{array}\right] \\ D=\left[\begin{array}{rrr} 7 & 3 \end{array}\right] & E=\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right] \\ F=\left[\begin{array}{lll} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \quad G=\left[\begin{array}{rrr} 5 & -3 & 10 \\ 6 & 1 & 0 \\ -5 & 2 & 2 \end{array}\right] \end{array}$$ Carry out the indicated algebraic operation, or explain why it cannot be performed. $$A B E$$

Step-by-Step Solution

Answer

The result of multiplying matrices \(A\), \(B\), and \(E\) is \(\begin{pmatrix} 13 \\ -7 \end{pmatrix}\)

Step 1: Verify Dimensions for Matrix Multiplication

To multiply matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, check if matrix multiplication is feasible: \(A\) is a \(2 \times 2\) matrix, \(B\) is a \(2 \times 3\) matrix, and \(E\) is a \(3 \times 1\) matrix. Thus, for \(AB\), we have \(2 \times 2\) and \(2 \times 3\), which is feasible. For the result \(ABE\), we need \(AB\) \(\text{(be a)}\) \(2 \times 3\) matrix and \(E\), a \(3 \times 1\), which is compatible.

Step 2: Compute Matrix AB

Calculate the product of matrices \(A\) and \(B\):\[AB = \begin{pmatrix} 2 & -5 \ 0 & 7 \end{pmatrix} \begin{pmatrix} 3 & \frac{1}{2} & 5 \ 1 & -1 & 3 \end{pmatrix}\]Multiply the first row of \(A\) with columns of \(B\):\[(2)(3) + (-5)(1), \quad (2)(\frac{1}{2}) + (-5)(-1), \quad (2)(5) + (-5)(3)\]\[(6) + (-5), \quad (1) + (5), \quad (10) - (15)\]The results are 1, 6, and -5 respectively.Now, multiply the second row of \(A\) with columns of \(B\):\[(0)(3) + (7)(1), \quad (0)(\frac{1}{2}) + (7)(-1), \quad (0)(5) + (7)(3)\]\[(0) + (7), \quad (0) - (7), \quad (0) + (21)\]The results are 7, -7, and 21 respectively.So, \(AB = \begin{pmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{pmatrix}\).

Step 3: Compute Matrix Product ABE

Using the resulting matrix from \(AB\), we now multiply by \(E\):\[\begin{pmatrix} 1 & 6 & -5 \ 7 & -7 & 21 \end{pmatrix}\begin{pmatrix} 1 \ 2 \ 0 \end{pmatrix}\]Multiply each row of \(AB\) by the column in \(E\):First row: \((1)(1) + (6)(2) + (-5)(0) = 1 + 12 = 13\)Second row: \((7)(1) + (-7)(2) + (21)(0) = 7 - 14 = -7\)Hence, \(ABE = \begin{pmatrix} 13 \ -7 \end{pmatrix}\).

Key Concepts

Matrix Dimensions

In matrix multiplication, understanding matrix dimensions is crucial. Each matrix is defined by its number of rows and columns. These dimensions determine compatibility for matrix operations.
For two matrices, let's call them matrix A and matrix B, to be multiplied, the number of columns in matrix A must be equal to the number of rows in matrix B. This forms the foundation for determining if matrix multiplication is possible.
For instance:

• Matrix A with dimensions \( m \times n \) can only be multiplied by matrix B with dimensions \( n \times p \).
• The resulting matrix from this multiplication will have dimensions \( m \times p \).

Thus, if you were given two matrices like in our example with A as a \( 2 \times 2 \) matrix and B as a \( 2 \times 3 \) matrix, they can be multiplied. However, always ensure to double-check these dimensions before attempting multiplication.

Matrices

Matrices are fundamental objects in linear algebra. They are rectangular arrays of numbers, symbols, or expressions, arranged in rows and columns.
They can be of any size, pattern, or variety, playing a critical role in mathematical computations across various fields such as physics, computer science, and engineering.
There are different types of matrices defined by specific properties:

• Square Matrix: A matrix with the same number of rows and columns (e.g., a \(3 \times 3\) matrix).
• Rectangular Matrix: A matrix with different numbers of rows and columns (e.g., a \(2 \times 3\) matrix).
• Identity Matrix: A square matrix with 1s on the diagonal and 0s elsewhere. It acts like the number 1 in matrix multiplication.
• Zero Matrix: A matrix with all elements being zero.

Understanding matrices and their types can help you identify the right operations and applications for each in mathematical problems.

Matrix Algebra

Matrix algebra involves operations on matrices, including addition, subtraction, multiplication, and finding inverses, among others. These operations follow specific rules, often parallel to algebraic rules, but with their unique characteristics.
Let's explore some basic operations:

• Addition and Subtraction: Matrices can only be added or subtracted if they have the same dimensions. This operation involves adding or subtracting corresponding entries.
• Multiplication: Beyond verifying dimensions for multiplication, the operation involves taking rows from the first matrix and columns from the second, multiplying corresponding entries, and summing these products.
• Inverse: Only square matrices have an inverse, and a matrix can only be invertible if its determinant is non-zero. The inverse matrix is such that when multiplied with the original matrix, results in an identity matrix.

By mastering these concepts, you can unlock vast applications of matrices in solving linear equations, transforming geometrical forms, and even in computer algorithms. Each operation is a building block in the larger structure of matrix algebra applied in different theoretical and practical problems.