Problem 54 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

\(AB = \left[ \begin{array}{cc} -2 & 0 \\ -2 & -7 \end{array} \right]\), \(BA = \left[ \begin{array}{ccc} -10 & 4 & -6 \\ -9 & 2 & -11 \\ 4 & -2 & 1 \end{array} \right]\).

Step 1: Checking Matrix Dimensions for Multiplication

For matrix multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(A\) is a \(2 \times 3\) matrix and matrix \(B\) is a \(3 \times 2\) matrix. Thus, \(AB\) is possible and will result in a \(2 \times 2\) matrix.

Step 2: Finding the Product \(AB\)

To find \(AB\), multiply each element of rows in \(A\) by each corresponding element of columns in \(B\) and calculate their sums. - First entry \((1,1)\): \((-1 \times 2) + (0 \times 5) + (-2 \times 0) = -2\)- Second entry \((1,2)\): \((-1 \times -2) + (0 \times -1) + (-2 \times 1) = 0\)- Third entry \((2,1)\): \((4 \times 2) + (-2 \times 5) + (1 \times 0) = -2\)- Fourth entry \((2,2)\): \((4 \times -2) + (-2 \times -1) + (1 \times 1) = -7\)Thus, the product \(AB\) is: \[AB = \left[ \begin{array}{cc}-2 & 0 \-2 & -7\end{array} \right]\].

Step 3: Checking Matrix Dimensions for \(BA\)

Now, let's check if \(BA\) is possible. Matrix \(B\) is \(3 \times 2\) and matrix \(A\) is \(2 \times 3\). The resulting matrix \(BA\) will be a \(3 \times 3\) matrix.

Step 4: Finding the Product \(BA\)

Multiply each element of rows in \(B\) by each corresponding element of columns in \(A\) and calculate their sums. - First entry \((1,1)\): \((2 \times -1) + (-2 \times 4) = -10\)- Second entry \((1,2)\): \((2 \times 0) + (-2 \times -2) = 4\)- Third entry \((1,3)\): \((2 \times -2) + (-2 \times 1) = -6\)- Fourth entry \((2,1)\): \((5 \times -1) + (-1 \times 4) = -9\)- Fifth entry \((2,2)\): \((5 \times 0) + (-1 \times -2) = 2\)- Sixth entry \((2,3)\): \((5 \times -2) + (-1 \times 1) = -11\)- Seventh entry \((3,1)\): \((0 \times -1) + (1 \times 4) = 4\)- Eighth entry \((3,2)\): \((0 \times 0) + (1 \times -2) = -2\)- Ninth entry \((3,3)\): \((0 \times -2) + (1 \times 1) = 1\)Thus, the product \(BA\) is: \[BA = \left[ \begin{array}{ccc}-10 & 4 & -6 \-9 & 2 & -11 \4 & -2 & 1\end{array} \right]\].

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial in the world of linear algebra, especially when it comes to matrix multiplication. A matrix is essentially a rectangular array of numbers, and its dimensions are defined by the number of rows and columns it has. For instance, a matrix with 3 rows and 2 columns is referred to as a \(3 \times 2\) matrix. This notation is important because it dictates how matrices can be multiplied.

When multiplying two matrices, the dimensions must allow the operation to be defined. Specifically, matrix multiplication requires that the number of columns in the first matrix matches the number of rows in the second matrix. This condition ensures that each element in a row of the first matrix has a corresponding element in a column of the second matrix, which forms the basis for multiplication. In our original exercise, matrix \(A\) is \(2 \times 3\) and matrix \(B\) is \(3 \times 2\). Thus, multiplying \(A\) by \(B\) (\(AB\)) is possible because the number of columns in \(A\) matches the number of rows in \(B\). The product will result in a new \(2 \times 2\) matrix.

Understanding these dimensions is important for ensuring that any matrix operations conducted are mathematically valid.

Product of Matrices

The product of matrices is found through a process called matrix multiplication. This requires multiplying rows from the first matrix by columns of the second matrix, following specific rules of arithmetic operations.

Consider matrix multiplication with two matrices: \(A\) and \(B\). To compute their product \(AB\), you perform an operation called the 'dot product' for each element in the resulting matrix. You take the corresponding elements of a row in matrix \(A\), multiply them by the corresponding elements from a column of matrix \(B\), and then sum those products. Repeat this process for each row and column pair to fill the resulting matrix.

In the provided example, the product \(AB\) results in the following matrix:
\[AB = \begin{bmatrix}-2 & 0 \-2 & -7\end{bmatrix}\]

Matrix multiplication is not commutative, meaning \(AB\) may not equal \(BA\). Each configuration can give a different dimension and result, as seen in the exercise.

Matrix Operations

Matrix operations form the foundation of numerous applications in science and engineering, and among these, matrix multiplication is a pivotal operation. Understanding how to perform these operations allows for more advanced tasks such as solving linear equations, transforming geometric data, or even encoding digital signals.

Apart from multiplication, which combines the entries of matrices in a rule-based manner, other operations include addition and subtraction, which are relatively straightforward. These operations require matrices to be of the same dimension to add or subtract their corresponding elements directly. However, multiplication is more intricate and necessitates careful attention to matrix dimensions and the order of multiplication.

In the context of the exercise, matrix \(B\) multiplied by matrix \(A\) (\(BA\)) results in the matrix:
\[BA = \begin{bmatrix}-10 & 4 & -6 \-9 & 2 & -11 \4 & -2 & 1\end{bmatrix}\]

This result shows the power and complexity of matrix operations, highlighting the versatility of matrices in representing and solving systems of equations and more complex mathematical models. Thus, a strong grasp of matrix operations is a fundamental skill in mathematical problem-solving.