Problem 42 - College Algebra with Modeling and Visualization

Question

If possible, find \(A B\) and \(B A\). $$A=\left[\begin{array}{rrr}1 & -2 & 5 \\\1 & 0 & -2 \\\1 & 3 & 2\end{array}\right], \quad B=\left[\begin{array}{rrr}-1 & 4 & 2 \\\\-3 & 0 & 1 \\\5 & 1 & 0\end{array}\right]$$

Step-by-Step Solution

Answer

Matrix AB is \(\begin{bmatrix} 30 & 9 & 0 \\ -11 & 2 & 2 \\ 0 & 6 & 5 \end{bmatrix}\), and matrix BA is \(\begin{bmatrix} 5 & 8 & -9 \\ -2 & 9 & -13 \\ 6 & -10 & 23 \end{bmatrix}\).

Step 1: Verify Dimensions Compatibility

To multiply two matrices \(A\) and \(B\), the number of columns in matrix \(A\) must equal the number of rows in matrix \(B\). Matrix \(A\) is of size \(3 \times 3\) and matrix \(B\) is of size \(3 \times 3\). Since the number of columns in \(A\) (3 columns) equals the number of rows in \(B\) (3 rows), the multiplication \(A \times B\) is possible. Similarly, since both are \(3 \times 3\) matrices, \(B \times A\) is also possible.

Step 2: Calculate Matrix AB

To find the product \(AB\), take each row of \(A\) and multiply it by each column of \(B\), summing the products. Calculate each element:- \((AB)_{11} = 1 \times (-1) + (-2) \times (-3) + 5 \times 5 = -1 + 6 + 25 = 30\)- \((AB)_{12} = 1 \times 4 + (-2) \times 0 + 5 \times 1 = 4 + 0 + 5 = 9\)- \((AB)_{13} = 1 \times 2 + (-2) \times 1 + 5 \times 0 = 2 - 2 + 0 = 0\)- \((AB)_{21} = 1 \times (-1) + 0 \times (-3) + (-2) \times 5 = -1 + 0 - 10 = -11\)- \((AB)_{22} = 1 \times 4 + 0 \times 0 + (-2) \times 1 = 4 + 0 - 2 = 2\)- \((AB)_{23} = 1 \times 2 + 0 \times 1 + (-2) \times 0 = 2 + 0 + 0 = 2\)- \((AB)_{31} = 1 \times (-1) + 3 \times (-3) + 2 \times 5 = -1 - 9 + 10 = 0\)- \((AB)_{32} = 1 \times 4 + 3 \times 0 + 2 \times 1 = 4 + 0 + 2 = 6\)- \((AB)_{33} = 1 \times 2 + 3 \times 1 + 2 \times 0 = 2 + 3 + 0 = 5\)Thus, \(AB = \begin{bmatrix} 30 & 9 & 0 \-11 & 2 & 2 \ 0 & 6 & 5 \end{bmatrix}\).

Step 3: Calculate Matrix BA

Now, calculate \(BA\) by taking each row of \(B\) and multiplying it by each column of \(A\), summing the products.Calculate each element:- \((BA)_{11} = (-1) \times 1 + 4 \times 1 + 2 \times 1 = -1 + 4 + 2 = 5\)- \((BA)_{12} = (-1) \times (-2) + 4 \times 0 + 2 \times 3 = 2 + 0 + 6 = 8\)- \((BA)_{13} = (-1) \times 5 + 4 \times (-2) + 2 \times 2 = -5 - 8 + 4 = -9\)- \((BA)_{21} = (-3) \times 1 + 0 \times 1 + 1 \times 1 = -3 + 0 + 1 = -2\)- \((BA)_{22} = (-3) \times (-2) + 0 \times 0 + 1 \times 3 = 6 + 0 + 3 = 9\)- \((BA)_{23} = (-3) \times 5 + 0 \times (-2) + 1 \times 2 = -15 + 0 + 2 = -13\)- \((BA)_{31} = 5 \times 1 + 1 \times 1 + 0 \times 1 = 5 + 1 + 0 = 6\)- \((BA)_{32} = 5 \times (-2) + 1 \times 0 + 0 \times 3 = -10 + 0 + 0 = -10\)- \((BA)_{33} = 5 \times 5 + 1 \times (-2) + 0 \times 2 = 25 - 2 + 0 = 23\)Thus, \(BA = \begin{bmatrix} 5 & 8 & -9 \-2 & 9 & -13 \ 6 & -10 & 23 \end{bmatrix}\).

Key Concepts

Matrix Algebra

Matrix algebra is a branch of mathematics that deals with mathematical operations on matrices. This is a foundational concept in higher mathematics and underpins various applications in science and engineering. Matrices are essentially rectangular arrays of numbers, symbols, or expressions arranged in rows and columns. Understanding matrix algebra gives us the power to compactly represent and solve systems of equations.

• In matrix algebra, matrices can be added, subtracted, and multiplied—each operation following specific rules.
• Matrix addition and subtraction require matrices of the same dimensions, whereas multiplication requires the number of columns in the first matrix to match the number of rows in the second matrix.

The multiplication of matrices, unlike numbers, is not commutative, which means that the order of multiplication matters. This is why in many exercises, such as the one provided, you can find both products like \(A \times B\) and \(B \times A\) calculated separately. Ensuring that you have the correct dimensions is crucial before performing matrix operations.

Linear Algebra

Linear algebra is a crucial field of mathematics that focuses on linear equations, linear mappings, and their representations in vector spaces and through matrices. It plays a vital role in various disciplines such as engineering, physics, computer science, economics, and statistics. In the context of linear algebra, matrices are used to solve systems of linear equations and to perform transformations of linear spaces.

• Linear algebra provides the tools to manipulate vector spaces and solve equations that describe linear relationships.
• Matrices come into play as powerful representations of these ideas, allowing for systematic solutions to complex problems.
• Determinants and eigenvalues are other significant concepts within linear algebra, often explored through matrix operations.

Understanding the principles of linear algebra helps to identify the properties and behaviors of different mathematical systems, especially when stemming from real-world problems.

Matrix Products

Matrix products involve the multiplication of two matrices to create a new matrix. This operation is central to many mathematical computations. When we say we are finding the product of two matrices \(A\) and \(B\), each element of the resulting matrix is computed by taking the dot product of the corresponding row from matrix \(A\) and column from matrix \(B\).Here is how matrix multiplication works:

• Consider two matrices, \(A\) and \(B\). To multiply these, the number of columns in \(A\) must match the number of rows in \(B\).
• The element in the ith row and jth column of the resulting matrix is obtained by multiplying the elements of the ith row of \(A\) with the corresponding elements in the jth column of \(B\), and then summing these products.
• Matrix multiplication is associative but not commutative, meaning \( (A \times B) \times C = A \times (B \times C) \), but \( A \times B e B \times A \) in general.

In the example exercise, both \(AB\) and \(BA\) were calculated, illustrating that changing the order of multiplication changes the resulting matrix. This understanding is crucial for many applications in scientific computing and data processing.