Problem 1 - Precalculus: Functions and Graphs

Question

Exer. \(1-2:\) Show that \(B\) is the inverse of \(A\) $$A=\left[\begin{array}{ll} 5 & 7 \\ 2 & 3 \end{array}\right], \quad B=\left[\begin{array}{rr} 3 & -7 \\ -2 & 5 \end{array}\right]$$

Step-by-Step Solution

Answer

Matrix \(B\) is the inverse of matrix \(A\) because \(AB = BA = I\).

Step 1: Define Inverse Matrices

Two matrices, \(A\) and \(B\), are inverses of each other if their product is the identity matrix: \(AB = I\) and \(BA = I\). The identity matrix \(I\) for a 2x2 matrix is \(\left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\).

Step 2: Multiply Matrices A and B

Calculate the product \(AB\): \[AB = \left[\begin{array}{cc} 5 & 7 \ 2 & 3 \end{array}\right] \left[\begin{array}{rr} 3 & -7 \ -2 & 5 \end{array}\right] = \left[\begin{array}{cc} (5 \times 3 + 7 \times (-2)) & (5 \times (-7) + 7 \times 5) \ (2 \times 3 + 3 \times (-2)) & (2 \times (-7) + 3 \times 5) \end{array}\right]\]Calculate each element: - Top-left: \(5 \cdot 3 + 7 \cdot (-2) = 15 - 14 = 1\) - Top-right: \(5 \cdot (-7) + 7 \cdot 5 = -35 + 35 = 0\) - Bottom-left: \(2 \cdot 3 + 3 \cdot (-2) = 6 - 6 = 0\) - Bottom-right: \(2 \cdot (-7) + 3 \cdot 5 = -14 + 15 = 1\) Thus, \(AB = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\).

Step 3: Multiply Matrices B and A

Calculate the product \(BA\): \[BA = \left[\begin{array}{rr} 3 & -7 \ -2 & 5 \end{array}\right] \left[\begin{array}{cc} 5 & 7 \ 2 & 3 \end{array}\right] = \left[\begin{array}{cc} (3 \times 5 + (-7) \times 2) & (3 \times 7 + (-7) \times 3) \ (-2 \times 5 + 5 \times 2) & (-2 \times 7 + 5 \times 3) \end{array}\right]\]Calculate each element: - Top-left: \(3 \cdot 5 + (-7) \cdot 2 = 15 - 14 = 1\) - Top-right: \(3 \cdot 7 + (-7) \cdot 3 = 21 - 21 = 0\) - Bottom-left: \((-2) \cdot 5 + 5 \cdot 2 = -10 + 10 = 0\) - Bottom-right: \((-2) \cdot 7 + 5 \cdot 3 = -14 + 15 = 1\) Thus, \(BA = \left[\begin{array}{cc} 1 & 0 \ 0 & 1 \end{array}\right]\).

Step 4: Conclusion

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

Key Concepts

Matrix Multiplication

Matrix multiplication is an essential concept in linear algebra. It involves combining two matrices to produce a third matrix. Unlike regular multiplication, order matters in matrix multiplication. Given matrices \(A\) and \(B\), their product \(AB\) is not necessarily equal to \(BA\). Understanding this order is crucial for operations that involve inverses and transformations.

Matrix multiplication involves taking each element of a row in the first matrix and multiplying it by the corresponding element in a column of the second matrix. The products in each row-column positions are then summed to obtain a single element in the resulting matrix.

• Step 1: Take a row from the first matrix.
• Step 2: Take a column from the second matrix.
• Step 3: Multiply the corresponding elements and sum them.
• Step 4: Place this sum in the corresponding position in the resulting matrix.

Consider an example to illustrate this:Given matrices:\[A = \begin{bmatrix} 5 & 7 \ 2 & 3 \end{bmatrix}, B = \begin{bmatrix} 3 & -7 \ -2 & 5 \end{bmatrix}\]The product \(AB\) is calculated as follows:\[AB = \begin{bmatrix} 5 \times 3 + 7 \times (-2) & 5 \times (-7) + 7 \times 5 \ 2 \times 3 + 3 \times (-2) & 2 \times (-7) + 3 \times 5 \end{bmatrix}= \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]

Identity Matrix

The identity matrix plays a pivotal role in matrix operations, much like the number 1 does in arithmetic. When any matrix is multiplied by the identity matrix, the result is the original matrix. This property is vital in confirming whether two matrices are inverses of one another.

For a \(2\times2\) matrix, the identity matrix is:\[I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]Notice how the diagonal from the top left to the bottom right contains all ones, while all other elements are zeroes. This specific arrangement guarantees that when a matrix is multiplied by the identity matrix, each element keeps its original position and value.

• Multiplicative Identity: \(AI = IA = A\)
• Used to verify inverses: \(AB = I\) and \(BA = I\) means \(B\) is the inverse of \(A\).

The identity matrix confirms the inverse relationship when multiplying two potential inverse matrices, as shown in the example:Given:\[AB = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}, BA = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}\]suggesting that both products returned the identity matrix, confirming the inverse relationship.

2x2 Matrices

A \(2x2\) matrix is a rectangular array with two rows and two columns. These matrices are straightforward yet powerful as they lay the foundation for understanding more advanced matrix concepts.

Every element within a \(2x2\) matrix is crucial in operations like addition, multiplication, and finding inverses. For a matrix \[\begin{bmatrix} a & b \ c & d \end{bmatrix}\], understand that the specific values directly affect calculations.

• Operations: Addition, Subtraction, Multiplication, and Finding Inverses.
• Special Properties: For non-invertible matrices, the determinant is \(ad - bc = 0\).

In the context of inverses:To find the inverse of a \(2x2\) matrix\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\],ensure the determinant \(ad - bc eq 0\) and use:\[\text{Inverse} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]In our example, the matrices \(A\) and \(B\) demonstrate such properties and calculations, showing \(B\) is indeed the inverse of \(A\) since \(A \times B\) and \(B \times A\) both yield the identity matrix.