Problem 23 - Advanced Engineering Mathematics

Question

$$ \begin{aligned} &\text { If } \mathbf{A}=\left(\begin{array}{ll} 3 & 4 \\ 8 & 1 \end{array}\right) \text { and } \mathbf{B}=\left(\begin{array}{rr} 5 & 10 \\ -2 & -5 \end{array}\right), \text { find }(\mathbf{a})(\mathbf{A B})^{T} \text { , }\\\ &\text { (b) } \mathbf{B}^{T} \mathbf{A}^{T} \end{aligned} $$

Step-by-Step Solution

Answer

Both \((\mathbf{A B})^T\) and \(\mathbf{B}^T \mathbf{A}^T\) equal \(\begin{pmatrix} 7 & 38 \\ 10 & 75 \end{pmatrix}\).

Step 1: Compute the Product \( \mathbf{A} \mathbf{B} \)

To find \( (\mathbf{A B})^T \), first compute the matrix product \( \mathbf{A} \mathbf{B} \).\[\mathbf{A} = \begin{pmatrix} 3 & 4 \ 8 & 1 \end{pmatrix}, \ \mathbf{B} = \begin{pmatrix} 5 & 10 \ -2 & -5 \end{pmatrix}\]Use the formula:\[\mathbf{A} \mathbf{B} = \begin{pmatrix} 3 \cdot 5 + 4 \cdot (-2) & 3 \cdot 10 + 4 \cdot (-5) \ 8 \cdot 5 + 1 \cdot (-2) & 8 \cdot 10 + 1 \cdot (-5) \end{pmatrix}\]\[\mathbf{A} \mathbf{B} = \begin{pmatrix} 15 - 8 & 30 - 20 \ 40 - 2 & 80 - 5 \end{pmatrix} = \begin{pmatrix} 7 & 10 \ 38 & 75 \end{pmatrix}\]

Step 2: Transpose \( \mathbf{A} \mathbf{B} \)

Transpose the result from Step 1 to find \( (\mathbf{A B})^T \).\[(\mathbf{A B})^T = \begin{pmatrix} 7 & 38 \ 10 & 75 \end{pmatrix}\]

Step 3: Compute the Transpose of Matrices \( \mathbf{B}^T \) and \( \mathbf{A}^T \)

Find \( \mathbf{B}^T \) and \( \mathbf{A}^T \) by transposing each matrix.\[\mathbf{B}^T = \begin{pmatrix} 5 & -2 \ 10 & -5 \end{pmatrix}, \ \mathbf{A}^T = \begin{pmatrix} 3 & 8 \ 4 & 1 \end{pmatrix}\]

Step 4: Compute \( \mathbf{B}^T \mathbf{A}^T \)

Multiply the transposed matrices to find \( \mathbf{B}^T \mathbf{A}^T \).\[\mathbf{B}^T \mathbf{A}^T = \begin{pmatrix} 5 & -2 \ 10 & -5 \end{pmatrix} \begin{pmatrix} 3 & 8 \ 4 & 1 \end{pmatrix}\]Using the multiplication formula:\[\mathbf{B}^T \mathbf{A}^T = \begin{pmatrix} 5 \cdot 3 + (-2) \cdot 4 & 5 \cdot 8 + (-2) \cdot 1 \ 10 \cdot 3 + (-5) \cdot 4 & 10 \cdot 8 + (-5) \cdot 1 \end{pmatrix}\]\[\mathbf{B}^T \mathbf{A}^T = \begin{pmatrix} 15 - 8 & 40 - 2 \ 30 - 20 & 80 - 5 \end{pmatrix} = \begin{pmatrix} 7 & 38 \ 10 & 75 \end{pmatrix}\]

Step 5: Verify \( (\mathbf{A B})^T = \mathbf{B}^T \mathbf{A}^T \)

Observe that the result of \( (\mathbf{A B})^T \) from Step 2 is equal to the result of \( \mathbf{B}^T \mathbf{A}^T \) from Step 4: \[\begin{pmatrix} 7 & 38 \ 10 & 75 \end{pmatrix} = \begin{pmatrix} 7 & 38 \ 10 & 75 \end{pmatrix}\]This confirms that the calculations are correct.

Key Concepts

Matrix Multiplication

Matrix multiplication involves taking two matrices and creating a new one by performing mathematical operations. It is crucial to remember that order matters in matrix multiplication. You can multiply two matrices only when the number of columns in the first matrix is equal to the number of rows in the second matrix.
To multiply two matrices, consider the following steps:

• Take the rows of the first matrix (let's call it Matrix A).
• Take the columns of the second matrix (call this Matrix B).
• Multiply each element of the row from Matrix A with the corresponding element of the column from Matrix B.
• Add up all these products to get an element of the new matrix.

This process repeats for every row and column combination to complete the new matrix, known as the product matrix. It's a bit like finding the dot product between each pair of rows and columns from the two matrices.

Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations and linear maps and their representations through matrices and vector spaces. As a foundation for understanding matrices, it helps in addressing problems involving multiple variables.
Key concepts in linear algebra include:

• Scalar multiplication: This involves multiplying every entry of a matrix by a scalar (a real number).
• Vector spaces: These spaces are composed of vectors, which are objects that can be added together and multiplied by scalars.
• Matrix transformations: These represent linear transformations of vector spaces and are the building blocks for understanding more complex systems.

Linear algebra finds applications in various fields such as science, engineering, computer graphics, and machine learning, where transformations and manipulations of data structures are needed for analysis and problem-solving.

Matrix Operations

Matrix operations include several different calculations you can perform on matrices, each with specific rules and requirements.
Here are some of the most essential operations:

• Addition/Subtraction: You can only add or subtract matrices with the same dimensions. The operation is done element-wise, adding or subtracting each respective element.
• Transpose: Transposing a matrix involves flipping it over its diagonal. The rows become columns and vice versa. The transpose of matrix A is denoted as A^T.
• Multiplication: We have previously discussed matrix multiplication, but it's essential to note that it is not commutative (i.e., AB ≠ BA unless under specific conditions).
• Inverse: Not all matrices have an inverse, but those that do are called invertible or non-singular. The inverse of matrix A is denoted as A^-1 and has the property that AA^-1 = I, where I is the identity matrix.

Understanding these operations is fundamental to solving systems of equations, transforming coordinate systems, and other applications within mathematics and applied sciences.