Problem 53 - College Algebra with Corequisite Support

Question

For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution. \(A=\left[\begin{array}{rrr}-2 & 0 & 9 \\ 1 & 8 & -3 \\ 0.5 & 4 & 5\end{array}\right], B=\left[\begin{array}{rrr}0.5 & 3 & 0 \\ -4 & 1 & 6 \\\ 8 & 7 & 2\end{array}\right], C=\left[\begin{array}{lll}1 & 0 & 1 \\ 0 & 1 & 0 \\\ 1 & 0 & 1\end{array}\right]\) \(B C\)

Step-by-Step Solution

Answer

Matrix multiplication \(BC\) is possible, and \(BC = \begin{bmatrix}0.5 & 3 & 0.5 \\ 2 & 1 & 2 \\ 10 & 7 & 10\end{bmatrix}.\)

Step 1: Identify the matrices involved

We are given matrices \(B\) and \(C\). First, let's write down their dimensions. Matrix \(B\) is a \(3 \times 3\) matrix, and matrix \(C\) is also a \(3 \times 3\) matrix.

Step 2: Check if multiplication is possible

Matrix multiplication is possible when the number of columns in the first matrix equals the number of rows in the second matrix. Both \(B\) and \(C\) are \(3 \times 3\) matrices, so the multiplication \(B \cdot C\) is possible.

Step 3: Multiply the matrices

To find \(BC\), we multiply the rows of \(B\) by the columns of \(C\). Calculate each element of the resulting matrix as follows:\[\begin{align*}BC_{11} & = 0.5 \cdot 1 + 3 \cdot 0 + 0 \cdot 1 = 0.5, \BC_{12} & = 0.5 \cdot 0 + 3 \cdot 1 + 0 \cdot 0 = 3, \BC_{13} & = 0.5 \cdot 1 + 3 \cdot 0 + 0 \cdot 1 = 0.5, \BC_{21} & = (-4) \cdot 1 + 1 \cdot 0 + 6 \cdot 1 = 2, \BC_{22} & = (-4) \cdot 0 + 1 \cdot 1 + 6 \cdot 0 = 1, \BC_{23} & = (-4) \cdot 1 + 1 \cdot 0 + 6 \cdot 1 = 2, \BC_{31} & = 8 \cdot 1 + 7 \cdot 0 + 2 \cdot 1 = 10, \BC_{32} & = 8 \cdot 0 + 7 \cdot 1 + 2 \cdot 0 = 7, \BC_{33} & = 8 \cdot 1 + 7 \cdot 0 + 2 \cdot 1 = 10.\end{align*}\]

Step 4: Write the resulting matrix

The product matrix \(BC\) is given by:\[BC = \begin{bmatrix}0.5 & 3 & 0.5 \ 2 & 1 & 2 \ 10 & 7 & 10\end{bmatrix}.\]

Step 5: Verify with a calculator

Use a calculator to input the matrices \(B\) and \(C\) and verify the multiplication result. Ensure the calculated matrix \(BC = \begin{bmatrix}0.5 & 3 & 0.5 \ 2 & 1 & 2 \ 10 & 7 & 10\end{bmatrix}\) matches the manual calculation.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is a fundamental aspect of working with matrices. Each matrix is defined by the number of rows (horizontal) and columns (vertical) it contains. For example, if a matrix has three rows and three columns, it is referred to as a 3x3 matrix. The dimensions of a matrix are crucial when performing operations like matrix multiplication. Matrix dimensions dictate whether two matrices can be multiplied. In our exercise, we see that both matrices B and C are 3x3 matrices. This means each has three rows and three columns. For matrix multiplication to be feasible, the number of columns in the first matrix must equal the number of rows in the second matrix. Here, since both matrices are 3x3, their dimensions align perfectly for multiplication, making the operation possible.

Matrix Operations

Matrix operations involve various methods of combining or transforming matrices to obtain desired results. One common operation is multiplication, which isn't as straightforward as multiplying individual numbers. Instead, matrix multiplication involves dot products between the rows of the first matrix and the columns of the second matrix.In the exercise, to compute \( BC \), we multiply each element from the rows of matrix \( B \) with corresponding elements in the columns of matrix \( C \), summing the products for each element in the resultant matrix. This detailed step ensures each entry of the resulting matrix is calculated precisely.- For the element in row one and column one of the product matrix \( BC \), you calculate it as \( 0.5 \times 1 + 3 \times 0 + 0 \times 1 = 0.5 \).- Continuing this process across all rows and columns constructs the complete product matrix. Overall, matrix operations like this require attention to each calculation, ensuring precision and accuracy in the resulting matrix.

Calculator Verification

When dealing with complex calculations like matrix multiplication, especially with larger matrices or more challenging entries, using a calculator can be a helpful tool for verification. Calculators, particularly scientific or graphing ones, often have capabilities designed to handle matrix operations.In our exercise scenario, once we manually calculate the matrix product \( BC \), verifying the result with a calculator ensures that no errors were made during manual steps. Here's how you can use a calculator for verification:

• Enter the matrix dimensions and values for both matrices \( B \) and \( C \) exactly as given.
• Follow the calculator's steps for multiplying matrices, typically by accessing a special function or menu designed for matrices.
• Compare the calculator’s output matrix with your manual calculation to confirm their accuracy.

Calculators can serve as a great backup, ensuring your understanding and steps are correct while providing confidence in your result.