Problem 39 - Precalculus: Mathematics for Calculus

Question

The matrices \(A, B,\) and \(C\) are defined as follows. $$A=\left[\begin{array}{rrr}0.3 & 1.1 & 2.4 \\\0.9 & -0.1 & 0.4 \\\\-0.7 & 0.3 & -0.5\end{array}\right] \quad B=\left[\begin{array}{cc}1.2 & -0.1 \\\0 & -0.5 \\\0.5 & -2.1\end{array}\right]$$ $$C=\left[\begin{array}{rrr}-0.2 & 0.2 & 0.1 \\\1.1 & 2.1 & -2.1\end{array}\right]$$ Use a graphing calculator to carry out the indicated algebraic operation, or explain why it cannot be performed. $$B C$$

Step-by-Step Solution

Answer

The product \( BC \) is possible and results in a \(3\times3\) matrix: \( \begin{bmatrix} -0.35 & 0.03 & 0.33 \\ -0.55 & -1.05 & 1.05 \\ -2.41 & -4.31 & 4.46 \end{bmatrix} \).

Step 1: Determine matrix dimensions

First, we need to identify the dimensions of matrices \(B\) and \(C\). Matrix \(B\) is a 3x2 matrix (3 rows and 2 columns), and matrix \(C\) is a 2x3 matrix (2 rows and 3 columns).

Step 2: Check matrix multiplication possibility

To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. Matrix \(B\) has 2 columns, and matrix \(C\) has 2 rows, which means the multiplication \(BC\) is possible.

Step 3: Multiply the matrices

Perform the multiplication of matrices \(B\) and \(C\). The result will be a 3x3 matrix, calculated as follows:\[BC = \begin{bmatrix} 1.2 & -0.1 \ 0 & -0.5 \ 0.5 & -2.1 \end{bmatrix}\begin{bmatrix} -0.2 & 0.2 & 0.1 \ 1.1 & 2.1 & -2.1 \end{bmatrix} = \begin{bmatrix} (1.2)(-0.2) + (-0.1)(1.1) & (1.2)(0.2) + (-0.1)(2.1) & (1.2)(0.1) + (-0.1)(-2.1) \ (0)(-0.2) + (-0.5)(1.1) & (0)(0.2) + (-0.5)(2.1) & (0)(0.1) + (-0.5)(-2.1) \ (0.5)(-0.2) + (-2.1)(1.1) & (0.5)(0.2) + (-2.1)(2.1) & (0.5)(0.1) + (-2.1)(-2.1) \end{bmatrix}\]Calculating each element gives us:\[\begin{bmatrix} -0.24 - 0.11 & 0.24 - 0.21 & 0.12 + 0.21 \ 0 - 0.55 & 0 - 1.05 & 0 + 1.05 \ -0.10 - 2.31 & 0.10 - 4.41 & 0.05 + 4.41 \end{bmatrix} = \begin{bmatrix} -0.35 & 0.03 & 0.33 \ -0.55 & -1.05 & 1.05 \ -2.41 & -4.31 & 4.46 \end{bmatrix}\].

Step 4: Verify the result

Verify by re-checking calculations to ensure each multiplication and addition was performed correctly step by step.

Key Concepts

Matrix Dimensions

Understanding matrix dimensions is crucial when dealing with matrices in algebraic operations. Matrix dimensions are denoted by the number of rows followed by the number of columns, typically written as "rows x columns". In our example, matrix \(B\) is a 3x2 matrix and matrix \(C\) is a 2x3 matrix.

It's important to note that when we multiply matrices, the number of columns in the first matrix must match the number of rows in the second matrix to make the operation possible. In this exercise, the multiplication of matrices \(B\) and \(C\) is feasible because the number of columns in \(B\), which is 2, is equal to the number of rows in \(C\). Remember, the resulting matrix after multiplication will have dimensions based on the outer dimensions of the two matrices being multiplied, which in this case is 3x3.

Algebraic Operations

Matrix multiplication is one of the fundamental algebraic operations involving matrices. However, unlike number multiplication, matrix multiplication involves taking rows from the first matrix and columns from the second matrix to compute the resulting matrix.

For each element of the resulting matrix, you multiply the corresponding elements in the row by the corresponding elements in the column and then sum these products. This process is repeated for each row and column pair in the matrices. In our step-by-step example, for the element in the first row and first column of the resulting matrix, we multiplied \(1.2\) by \(-0.2\) and \(-0.1\) by \(1.1\), then added these products to get \(-0.35\). Each element of the resulting matrix \(BC\) is similarly calculated.

Graphing Calculator

Using a graphing calculator can simplify complex calculations like matrix multiplication. These calculators can efficiently handle operations involving multiple steps and provide quick answers, freeing you from doing tedious arithmetic operations.

To use a graphing calculator for matrix multiplication, input the matrices as instructed by the calculator's guide, and use the matrix operations function to multiply them. This will instantly give you the resulting matrix. Remember, while a graphing calculator is a powerful tool, understanding the underlying steps is crucial in advancing your math skills. Therefore, practicing manual calculations alongside using technology is advisable.

Result Verification

After performing matrix multiplication, it's important to verify the result by double-checking each step of your calculations. This helps to ensure that no errors occurred during the multiplication and addition processes.

To verify, go through each element of the resulting matrix to confirm that each multiplication and subsequent addition was executed accurately. In our example, re-calculate the values for each element in the result matrix \(BC\) to ensure they match the correctly calculated values: \([-0.35, 0.03, 0.33], [-0.55, -1.05, 1.05], [-2.41, -4.31, 4.46]\). Verification is a critical step in mathematics as it helps reinforce accuracy and build confidence in handling matrix operations.