Problem 107 - Precalculus

Question

$$\begin{aligned} &\text { Verify that } A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\\\-c & a \end{array}\right] \text { is the inverse of }\\\ &A=\left[\begin{array}{ll}a & b \\\c & d\end{array}\right], \text { provided } a d-b c \neq 0 \end{aligned}$$

Step-by-Step Solution

Answer

The matrix is valid; the inverse satisfies the identity matrix condition.

Step 1: Understand the Matrix Inverse Definition

To verify that a matrix \( B \) is the inverse of a matrix \( A \), they must satisfy the condition \( A \times B = B \times A = I \), where \( I \) is the identity matrix of the same size.

Step 2: Set Up the Inverse Verification

Matrix \( A \) is given by \( A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \). The proposed inverse \( A^{-1} \) is \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \). Our goal is to show that their product is the identity matrix.

Step 3: Calculate \( A \times A^{-1} \)

Multiply matrix \( A \) by \( A^{-1} \): \[ A \times A^{-1} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \times \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix} \]Expand this product by following the matrix multiplication rule.

Step 4: Expand the Products

Compute the elements of the resulting matrix:- The element at position \((1, 1)\): \((a \times d) + (b \times -c) = ad - bc\)- The element at position \((1, 2)\): \((a \times -b) + (b \times a) = -ab + ab = 0\)- The element at position \((2, 1)\): \((c \times d) + (d \times -c) = cd - dc = 0\)- The element at position \((2, 2)\): \((c \times -b) + (d \times a) = -bc + ad\)

Step 5: Simplify the Result

Now simplify the resulting matrix:\[ A \times A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} ad-bc & 0 \ 0 & ad-bc \end{bmatrix} \] This simplifies further to:\[ A \times A^{-1} = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]because \( \frac{ad-bc}{ad-bc} = 1 \).

Step 6: Conclusion from the Simplified Result

Since the product \( A \times A^{-1} \) results in the identity matrix, \( A^{-1} \) is indeed the inverse of matrix \( A \). A similar process would show that \( A^{-1} \times A = I \), confirming the two-sided inverse property.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation where two matrices are multiplied together to produce another matrix. The number of columns in the first matrix must match the number of rows in the second matrix for the multiplication to be possible.

The process involves taking each element from the rows of the first matrix and matching it with elements from the columns of the second matrix. These matched elements are multiplied together and summed up to produce an individual element in the resulting matrix.

• Example: For a matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\) and its inverse, \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\), the multiplication \(A \times A^{-1}\) involves multiplying each row of \(A\) with each column of \(A^{-1}\).

Each calculation here is crucial to ensure the resulting matrix is accurate.

Identity Matrix

The identity matrix is a special type of matrix that acts like the number 1 in matrix algebra. When any matrix is multiplied by the identity matrix, the original matrix is unchanged.

For a square matrix of size \(n \times n\), the identity matrix is denoted as \(I_n\) and has 1s on the main diagonal and 0s elsewhere. For example, a 2x2 identity matrix looks like this:

• \(I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\)

Finding the inverse of a matrix involves proving that when it is multiplied by its inverse, the result is this identity matrix.

Both \(A \times A^{-1}\) and \(A^{-1} \times A\) need to equal the identity matrix to confirm that \(A^{-1}\) is indeed the inverse.

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements, and is a key factor used to determine whether a matrix has an inverse. The determinant for a 2x2 matrix \( \begin{bmatrix} a & b \ c & d \end{bmatrix} \) is calculated as follows:

• Determinant: \( \text{det}(A) = ad - bc \)

If the determinant is zero, the matrix does not have an inverse. This is because division by zero is undefined, making the inverse computation impossible.

In the exercise, the condition \( ad-bc eq 0 \) ensures that the matrix has an inverse.

When calculating the inverse, the formula \(A^{-1} = \frac{1}{ad-bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\) incorporates division by the determinant, emphasizing its role in matrix inversion.

Matrix Elements

Matrix elements are the individual items located in a matrix, usually denoted by positions \((i, j)\), where \(i\) represents the row and \(j\) represents the column. In a 2x2 matrix, the elements are located as follows:

• First row, first column: \(a\)
• First row, second column: \(b\)
• Second row, first column: \(c\)
• Second row, second column: \(d\)

When performing operations such as multiplication, each element plays a crucial role as they are combined according to specific rules to achieve the final matrix result.

Engaging with each element appropriately during matrix manipulation is essential to ensure accurate results, particularly when verifying inverses:

• Place every element correctly in calculations to derive results that lead to the identity matrix, as in the exercise.

Understanding how these elements interact is critical for mastering matrix operations.