Problem 13 - Algebra for College Students

Question

For Problems \(13-26\), compute \(A B\) and \(B A\). $$ A=\left[\begin{array}{ll} 1 & -1 \\ 2 & -2 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -4 \\ -1 & 2 \end{array}\right] $$

Step-by-Step Solution

Answer

\(AB = \left[\begin{array}{cc} 4 & -6 \\ 8 & -12 \end{array}\right]\) and \(BA = \left[\begin{array}{cc} -5 & 5 \\ 3 & -3 \end{array}\right]\).

Step 1: Understand Matrix Multiplication Requirements

Matrix multiplication is possible when the number of columns in the first matrix is equal to the number of rows in the second matrix. Both matrices, A and B, are 2x2, so we can compute both \(AB\) and \(BA\).

Step 2: Compute Matrix Product AB

To compute \(AB = A \cdot B\), multiply each element of the rows of matrix A with the corresponding element of the columns of matrix B and sum all the products for each position. The formula for each element \(C_{ij}\) of the resulting matrix C is given by:\[ C_{ij} = \sum_{k=1}^{n} A_{ik} \cdot B_{kj} \]For \(C_{11}\): \((1 \cdot 3) + (-1 \cdot -1) = 3 + 1 = 4\).For \(C_{12}\): \((1 \cdot -4) + (-1 \cdot 2) = -4 - 2 = -6\).For \(C_{21}\): \((2 \cdot 3) + (-2 \cdot -1) = 6 + 2 = 8\).For \(C_{22}\): \((2 \cdot -4) + (-2 \cdot 2) = -8 - 4 = -12\).Thus, \(AB = \left[\begin{array}{cc} 4 & -6 \ 8 & -12 \end{array}\right]\).

Step 3: Compute Matrix Product BA

To compute \(BA = B \cdot A\), multiply each element of the rows of matrix B with the corresponding element of the columns of matrix A and sum all the products for each position.For \(D_{11}\): \((3 \cdot 1) + (-4 \cdot 2) = 3 - 8 = -5\).For \(D_{12}\): \((3 \cdot -1) + (-4 \cdot -2) = -3 + 8 = 5\).For \(D_{21}\): \((-1 \cdot 1) + (2 \cdot 2) = -1 + 4 = 3\).For \(D_{22}\): \((-1 \cdot -1) + (2 \cdot -2) = 1 - 4 = -3\).Thus, \(BA = \left[\begin{array}{cc} -5 & 5 \ 3 & -3 \end{array}\right]\).

Key Concepts

Matrix Operations

Matrix operations are fundamental in mathematics, especially in linear algebra, and include activities like addition, subtraction, and multiplication. Any operation involving matrices can have certain rules. For multiplication, which is a key matrix operation, there are specific requirements:

• Each element in a matrix is identified by a unique position.
• The number of columns in the first matrix must equal the number of rows in the second matrix.

By meeting these requirements, we can engage in efficient matrix multiplication. It is a structured process that uses rows and columns intelligently.
To better understand, let's consider two 2x2 matrices A and B. These matrices can undergo various operations, with multiplication being one of the most crucial.

Matrix Product

The matrix product is obtained by multiplying matrices following certain rules. Let’s say we have two matrices A and B. The product of these matrices, denoted by AB, is a new matrix C.
To compute this:

• Take each row of the first matrix (A).
• Multiply it with each column of the second matrix (B).
• Sum each of these products to find the respective element in the product matrix C.

For example:

• To find the element in the first row, first column of matrix AB, multiply the first row of A by the first column of B and sum the results.
• Continue this process across all respective rows and columns.

The process differs slightly for each element of the resulting matrix, resulting in a new matrix representation of the original matrices' interactions.

Matrix Dimensions

Understanding matrix dimensions is crucial for conducting matrix operations. The dimensions of a matrix are given in the form of rows x columns.

• Matrix A, for example, may have dimensions 2x2, meaning it has 2 rows and 2 columns.
• This is the same for matrix B if it is used in this context.

In matrix multiplication, the dimensions tell us if and how we can multiply the matrices.
Using the earlier example, where both A and B are 2x2 matrices, checking the dimensions shows us they can be multiplied.

The multiplied matrices will also have dimensions determined by their rows and columns interaction during the operation process, typically resulting in another 2x2 matrix when both matrices are 2x2.