Problem 45 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rr}-3 & 1 \\\2 & -4\end{array}\right], \quad B=\left[\begin{array}{rrr}1 & 0 & -2 \\\\-4 & 8 & 1\end{array}\right]$$

Step-by-Step Solution

Answer

\( AB = \begin{bmatrix} -7 & 8 & 7 \\ 18 & -32 & -8 \end{bmatrix} \); \( BA \) is not possible.

Step 1: Check the Dimensions for Matrix Multiplication

For the multiplication of two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \( A \) is a \( 2 \times 2 \) matrix and matrix \( B \) is a \( 2 \times 3 \) matrix. Since the number of columns in \( A \) equals the number of rows in \( B \), the multiplication \( AB \) is possible. However, \( B \) is \( 2 \times 3 \) and \( A \) is \( 2 \times 2 \), so \( BA \) cannot be multiplied because the number of columns in \( B \) does not match the number of rows in \( A \).

Step 2: Calculate Matrix Product \( AB \)

Since \( A \) is \( 2 \times 2 \) and \( B \) is \( 2 \times 3 \), the resulting matrix \( AB \) will be \( 2 \times 3 \). To find this matrix, use the formula:\[(AB)_{ij} = \sum_{k=1}^{2} a_{ik}b_{kj}\], where \( i \) and \( j \) are the row and column indices of the resulting matrix, respectively.Calculate the entries:\[(AB)_{11} = (-3)(1) + (1)(-4) = -3 - 4 = -7\]\[(AB)_{12} = (-3)(0) + (1)(8) = 0 + 8 = 8\]\[(AB)_{13} = (-3)(-2) + (1)(1) = 6 + 1 = 7\]\[(AB)_{21} = (2)(1) + (-4)(-4) = 2 + 16 = 18\]\[(AB)_{22} = (2)(0) + (-4)(8) = 0 - 32 = -32\]\[(AB)_{23} = (2)(-2) + (-4)(1) = -4 - 4 = -8\]

Step 3: Write the Resultant Matrix \( AB \)

Based on the above calculations, the product matrix \( AB \) is:\[AB = \begin{bmatrix} -7 & 8 & 7 \ 18 & -32 & -8 \end{bmatrix}\]

Step 4: Reconfirm that \( BA \) is Not Possible

Matrix \( B \) cannot be multiplied by \( A \) as \( B \) is \( 2 \times 3 \) and \( A \) is \( 2 \times 2 \). The number of columns in \( B \) must equal the number of rows in \( A \) for multiplication to be possible, which is not the case.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is important in determining whether you can multiply two matrices. A matrix is simply a rectangular array of numbers, divided into rows and columns. The dimensions of a matrix are given as 'rows × columns'. For instance, a matrix with 2 rows and 3 columns is referred to as a 2 × 3 matrix.

When performing matrix multiplication, the critical rule is that the number of columns in the first matrix must equal the number of rows in the second matrix. Only if this condition is met can the matrices be multiplied. In our exercise, matrix A has dimensions 2 × 2, and matrix B has dimensions 2 × 3. Since the number of columns in A (which is 2) matches the number of rows in B (also 2), the product AB is possible to compute.

• The resulting matrix has dimensions from the outer sides: 2 rows (from A) and 3 columns (from B), so AB is a 2 × 3 matrix.
• If you were to attempt BA, you would need the number of columns in B (which is 3) to match the number of rows in A (which is 2), which doesn't work, hence BA is not possible.

Matrix Products

Matrix products involve a specific process of multiplication that produces a new matrix from two given matrices. To determine the elements of the resulting matrix, the elements of the rows of the first matrix are multiplied by the elements of the columns of the second matrix, and then summed.

For matrices A and B, the resulting product matrix AB can be calculated using the formula:

• \[(AB)_{ij} = \sum_{k=1}^{n} a_{ik} \, b_{kj}\], where:
• This formula represents the dot product of the i-th row of A and the j-th column of B.

For each element of AB, you iteratively multiply the corresponding elements from a row of A and a column of B, summing up all these products.
Let's highlight some calculations:

• To find the element in the first row and first column of AB, you multiply the first row of A by the first column of B: \((-3) \times 1 + 1 \times (-4) = -7\).
• Similarly, for the second row and third column: \(2 \times (-2) + (-4) \times 1 = -8\).

Algebraic Operations

In the context of matrices, algebraic operations refer to basic mathematical procedures like addition, subtraction, and multiplication that can be applied to matrices. Matrix multiplication is one of the most common algebraic operations and is not as straightforward as multiplying individual numbers.

Each element in the resulting matrix is derived from a sequence of multiplications and additions. It's important to follow the established rules for matrix arithmetic.

• Order matters in matrix operations: unlike regular number multiplication, AB does not necessarily equal BA.
• Both matrices must be conformable in size (having a compatible number of rows and columns) to be multiplied.

In our exercise, the criteria for multiplication (column count of A equaling row count of B) was met, allowing for the computation of AB, but not BA. Understanding these nuanced rules ensures correct execution of matrix algebra.