Problem 5 - Precalculus: Mathematics for Calculus

Question

Verifying the Inverse of a Matrix Calculate the products \(A B\) and \(B A\) to verify that \(B\) is the inverse of \(A\) \(A=\left[\begin{array}{rrr}1 & 3 & -1 \\ 1 & 4 & 0 \\ -1 & -3 & 2\end{array}\right] \quad B=\left[\begin{array}{rrr}8 & -3 & 4 \\ -2 & 1 & -1 \\\ 1 & 0 & 1\end{array}\right]\)

Step-by-Step Solution

Answer

Matrix \(B\) is the inverse of matrix \(A\) because both \(AB\) and \(BA\) equal the identity matrix.

Step 1: Matrix Multiplication Preparation

To verify if matrix \(B\) is the inverse of matrix \(A\), we need to compute \(AB\) and \(BA\). The products should equal the identity matrix \(I\), which is a 3x3 matrix given by \(\begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\).

Step 2: Compute Product AB

Calculate \( AB = A \cdot B \) by multiplying each row of \(A\) by each column of \(B\): - First row and first column: \(1\times8 + 3\times(-2) + (-1)\times1 = 8 - 6 - 1 = 1\).- First row and second column: \(1\times(-3) + 3\times1 + (-1)\times0 = -3 + 3 + 0 = 0\).- First row and third column: \(1\times4 + 3\times(-1) + (-1)\times1 = 4 - 3 - 1 = 0\).Similar calculations for the other rows and columns yield:\[ AB = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

Step 3: Compute Product BA

Next, calculate \( BA = B \cdot A \) by multiplying each row of \(B\) by each column of \(A\): - First row and first column: \(8\times1 + (-3)\times1 + 4\times(-1) = 8 - 3 - 4 = 1\).- First row and second column: \(8\times3 + (-3)\times4 + 4\times(-3) = 24 - 12 - 12 = 0\).- First row and third column: \(8\times(-1) + (-3)\times0 + 4\times2 = -8 + 0 + 8 = 0\).Proceed similarly for other rows and columns to find:\[ BA = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]

Step 4: Conclusion

Since both \(AB\) and \(BA\) equal the identity matrix \(I\), matrix \(B\) is confirmed to be the inverse of matrix \(A\).

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. It combines two matrices to create a new matrix, which represents a transformation involving both original matrices. It isn't as straightforward as number multiplication and follows unique rules:

• The number of columns in the first matrix must equal the number of rows in the second matrix.
• Elements are calculated by computing the dot product of rows and columns.

For instance, when multiplying matrix \(A\) by matrix \(B\), you take a row from \(A\) and a column from \(B\), multiply the corresponding elements, and sum the results to get a single element of the resulting matrix. As seen in the solution, each calculation builds upon these principles. This technique forms the basis for more complex operations in linear algebra and is crucial when working with inverse matrices.

Identity Matrix

The identity matrix is a special matrix in the world of mathematics. It's important because it acts as the "multiplicative identity" in matrix arithmetic, similar to how 1 acts for numbers. Simply put:

• When any matrix is multiplied by an identity matrix, it remains unchanged.
• In terms of notation, an identity matrix is often represented by \(I\) and contains 1's on its diagonal and 0's elsewhere.

For a 3x3 identity matrix, it looks like \[ \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix} \]In the original solution, the identity matrix is the goal for both products \(AB\) and \(BA\) when checking if \(B\) is the inverse of \(A\). Seeing this matrix as the result confirms that the operation between \(A\) and \(B\) mirrors each other out perfectly, ensuring the matrices are inverse.

Step-by-step Solution

When tackling complex problems, breaking them down into simple, manageable steps is effective. The step-by-step approach is a powerful method to ensure clarity and accuracy in solving matrix operations:

• First, preparation involves understanding required outcomes, such as aiming for the identity matrix.
• Next, systematically carry out matrix multiplication, confirming results at each stage.
• Always verify conclusions against the expected outcome, like the identity matrix in this example, to ensure correctness.

This organized tactic can be applied beyond matrix multiplication, benefiting various fields of study. In the given solution, each multiplication process is carefully broken down and calculated row by column, making it accessible to verify whether matrix \(B\) satisfies the condition to be \(A\)'s inverse.