Problem 18 - Advanced Engineering Mathematics

Question

If \(\mathbf{A}=\left(\begin{array}{rr}1 & 4 \\ 5 & 10 \\ 8 & 12\end{array}\right)\) and \(\mathbf{B}=\left(\begin{array}{rrr}-4 & 6 & -3 \\\ 1 & -3 & 2\end{array}\right)\), find (a) \(\mathbf{A B}\), (b) \(\mathbf{B A}\).

Step-by-Step Solution

Answer

\( \mathbf{A B} = \begin{pmatrix} 0 & -6 & 5 \\ -10 & 0 & -5 \\ -20 & 0 & -4 \end{pmatrix} \), \( \mathbf{B A} = \begin{pmatrix} 2 & 0 \\ 0 & -10 \end{pmatrix} \)."

Step 1: Identify Matrix Dimensions

Matrix \( \mathbf{A} \) is a \(3 \times 2\) matrix, and matrix \( \mathbf{B} \) is a \(2 \times 3\) matrix. Knowing their respective dimensions, the product \( \mathbf{A B} \) will result in a \(3 \times 3\) matrix, and the product \( \mathbf{B A} \) will result in a \(2 \times 2\) matrix.

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

To find \( \mathbf{A B} \), multiply the matrices using the definition of matrix multiplication. Each element \( c_{ij} \) of the resulting matrix is computed as the dot product of the \( i \)-th row of \( \mathbf{A} \) and the \( j \)-th column of \( \mathbf{B} \):\[\mathbf{A B} = \left(\begin{array}{rr}1 \times (-4) + 4 \times 1 & 1 \times 6 + 4 \times (-3) & 1 \times (-3) + 4 \times 2 \5 \times (-4) + 10 \times 1 & 5 \times 6 + 10 \times (-3) & 5 \times (-3) + 10 \times 2 \8 \times (-4) + 12 \times 1 & 8 \times 6 + 12 \times (-3) & 8 \times (-3) + 12 \times 2\end{array}\right)\]Simplifying calculations, we get:\[\mathbf{A B} = \left(\begin{array}{rrr} 0 & -6 & 5 \ -10 & 0 & -5 \ -20 & 0 & -4 \end{array}\right)\]

Step 3: Compute \( \mathbf{B A} \)

Similarly, calculate \( \mathbf{B A} \) by multiplying \( \mathbf{B} \) with \( \mathbf{A} \):\[\mathbf{B A} = \left(\begin{array}{rrr}(-4) \times 1 + 6 \times 5 + (-3) \times 8 & (-4) \times 4 + 6 \times 10 + (-3) \times 12 \1 \times 1 + (-3) \times 5 + 2 \times 8 & 1 \times 4 + (-3) \times 10 + 2 \times 12\end{array}\right)\]Simplify calculations:\[\mathbf{B A} = \left(\begin{array}{rr}2 & 0 \0 & -10\end{array}\right)\]

Step 4: Review Final Results

Now that the matrix multiplications are done, we summarize the results:\( \mathbf{A B} = \left(\begin{array}{rrr} 0 & -6 & 5 \ -10 & 0 & -5 \ -20 & 0 & -4 \end{array}\right) \) and\( \mathbf{B A} = \left(\begin{array}{rr} 2 & 0 \ 0 & -10 \end{array}\right) \).

Key Concepts

Matrix Dimensions

Matrix dimensions are crucial to understand before attempting matrix multiplication. Each matrix is defined by its number of rows and columns, often expressed as 'row by column'. For example, a matrix of dimensions \(3 \times 2\) has 3 rows and 2 columns. These dimensions dictate whether two matrices can be multiplied together.When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second. For instance, matrix \( \mathbf{A} \) has dimensions \(3 \times 2\) and matrix \( \mathbf{B} \) has dimensions \(2 \times 3\). Since the inner dimensions (2 in \( \mathbf{A} \) and 2 in \( \mathbf{B} \)) match, we can multiply these matrices.

• The result of this multiplication will have the dimensions of the outer numbers, meaning \( \mathbf{A B} \) will be a \(3 \times 3\) matrix.
• In contrast, if we multiply B by A, \( \mathbf{B A} \) will result in a \(2 \times 2\) matrix.

Recognizing and calculating matrix dimensions is the first step in ensuring that matrix multiplication is possible.

Dot Product

The dot product is an important element in matrix multiplication, forming the basis of calculating the elements in the resulting matrix. The dot product of two vectors is the sum of the products of corresponding elements.To compute the matrix multiplication of \( \mathbf{A} \) and \( \mathbf{B} \), we use the dot product between rows of A and columns of B. Specifically, each element \( c_{ij} \) in the product matrix is found by taking the dot product of the \(i\)-th row of \( \mathbf{A} \) with the \(j\)-th column of \( \mathbf{B} \).

• For example, for the first element in \( \mathbf{A B} \), \( c_{11} \), we calculate: \(1 \times (-4) + 4 \times 1\).
• This results in the value 0, filling the position in the product matrix.
• Each position in the resulting matrix is determined similarly, by continuing to pair each row from matrix \( \mathbf{A} \) with each column from matrix \( \mathbf{B} \).

Understanding and calculating dot products correctly are essential steps in performing accurate matrix multiplication.

Matrix Operations

Matrix operations involve a variety of mathematical processes, including addition, subtraction, scalar multiplication, and especially matrix multiplication. Among these, matrix multiplication is one of the more complex operations due to the specific rules required for its execution.

Rules of Matrix Multiplication:

Matrix multiplication does not follow the commutative property; that is, \( \mathbf{A B} eq \mathbf{B A} \) in general. The order of multiplication affects the resulting matrix.

• This is evident in our exercise, where \( \mathbf{A B} \) gives a \(3 \times 3\) matrix, while \( \mathbf{B A} \) results in a \(2 \times 2\) matrix.
• Complying with matrix dimension rules, the detailed step of multiplying involves calculating the dot product as discussed.

In solving matrix multiplication problems, breaking down into steps such as identifying matrix dimensions, performing necessary dot products, and respecting the non-commutative nature, will ensure correct outcomes. This detailed methodology is essential for mastering matrix operations.