Problem 46 - Precalculus

Question

determine whether \(B\) is the multiplicative inverse of \(A\) using \(A A^{-1}=I\) $$A=\left[\begin{array}{rr}7 & -9 \\\\-3 & 4\end{array}\right] \quad B=\left[\begin{array}{ll} 4 & 9 \\\3 & 7\end{array}\right]$$

Step-by-Step Solution

Answer

Yes, \( B \) is the inverse of \( A \) since \( AB = I \).

Step 1: Understand the Problem

We need to verify whether matrix \( B \) is the inverse of matrix \( A \). For \( B \) to be the inverse, the product \( AB \) (and \( BA \)) should equate to the identity matrix \( I \). For a 2x2 matrix, \( I \) has 1's on the diagonal and 0's elsewhere: \( I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

Step 2: Multiply Matrix A by Matrix B

Compute the product \( AB \) using matrix multiplication: \[AB = \begin{bmatrix} 7 & -9 \ -3 & 4 \end{bmatrix} \begin{bmatrix} 4 & 9 \ 3 & 7 \end{bmatrix}\]Calculate each entry:First row, first column: \((7 \times 4) + (-9 \times 3) = 28 - 27 = 1\)First row, second column: \((7 \times 9) + (-9 \times 7) = 63 - 63 = 0\)Second row, first column: \((-3 \times 4) + (4 \times 3) = -12 + 12 = 0\)Second row, second column: \((-3 \times 9) + (4 \times 7) = -27 + 28 = 1\)So, \( AB = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \).

Step 3: Verify Identity Matrix Property

Since the product \( AB = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \), it matches the identity matrix \( I \). For completion, you can check \( BA \) as well, but \( AB = I \) is sufficient to confirm that \( B \) is the inverse of \( A \).

Key Concepts

Identity Matrix

The identity matrix is a fundamental concept when working with matrices, especially in operations like determining matrix inversions. It serves as the "multiplicative identity" for matrix operations, similar to how the number 1 operates within the realm of real numbers.

For any matrix multiplication, when a matrix is multiplied by its identity matrix, the result is the original matrix. This can be represented as:

• If you have a matrix \(A\), then \(A \, I = A\).
• Similarly, \(I \, A = A\).

When identifying the inverse of a matrix, we rely on the identity matrix to verify our results. For a 2x2 matrix, the identity matrix is:
\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]
This matrix contains "1s" along the diagonal from the top left to the bottom right, and "0s" elsewhere. Checking if two matrices are inverses involves ensuring their multiplication results in the identity matrix.

Matrix Multiplication

Matrix multiplication is a key operation in linear algebra used to determine the product of two matrices. This process involves the dot product of the rows of the first matrix with the columns of the second matrix.

For two matrices \(A\) and \(B\) to be multiplied, the number of columns in matrix \(A\) must equal the number of rows in matrix \(B\). The resulting matrix product \(C\) will have dimensions of the rows of \(A\) by the columns of \(B\).

• In the exercise, matrix \(A\) is 2x2, and matrix \(B\) is 2x2, so they can be multiplied.
• Each element of the resulting matrix is calculated by a structured sum of products.

For instance, in calculating \(AB\), do the following steps:- For each element, take corresponding elements from the row of matrix \(A\) and column of matrix \(B\) and multiply them.- Add the results to get a single element in the resulting matrix.This systematic approach ensures you obtain the correct output, which can also be used to verify the identity matrix.

Inverse of a Matrix

An inverse matrix is a matrix that, when multiplied by its original matrix, yields the identity matrix. This is akin to finding a reciprocal in numerical operations, which when multiplied with the original number, results in 1.

To determine if a matrix \(B\) is the inverse of a matrix \(A\):

• Calculate the product \(AB\). If \(AB = I\) (where \(I\) is the identity matrix), then \(B\) is the inverse of \(A\).
• Ensure that \(BA = I\) as it confirms the reciprocal property, although checking the \(AB = I\) is typically sufficient.

If a 2x2 matrix \(A\) is:
\[A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\]The inverse, if it exists, is given by:
\[A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]This formula is only valid when the determinant \(ad-bc\) is non-zero. A zero determinant implies that the inverse does not exist, which is important for understanding when two matrices cannot be inverses.