Problem 47 - Algebra and Trigonometry

Question

Find \((A B)^{-1}, A^{-1} B^{-1},\) and \(B^{-1} A^{-1} .\) What do you observe? $$ A=\left[\begin{array}{ll} {2} & {1} \\ {3} & {1} \end{array}\right] \quad B=\left[\begin{array}{ll} {4} & {7} \\ {1} & {2} \end{array}\right] $$

Step-by-Step Solution

Answer

The result is: \(A^{-1}\) = \[\[-1, 1]\[3, -2]]\], \(B^{-1}\) = \[\[-2, 7]\[1, -4]]\], \((A B)^{-1}\) = \[\[-29, 21]\[7, -6]]\], \(A^{-1} B^{-1}\) = \[\[-8, 27]\[2, -7]]\] and \(B^{-1} A^{-1}\) = \[\[-8, 7]\[27, -22]]\]. Note that generally \(A^{-1} B^{-1}\) is not equal to \(B^{-1} A^{-1}\).

Step 1: Compute A Inverse

First we calculate the inverse of the matrix \(A\). The inverse of a 2x2 matrix \[\[a, c]\[b, d]]\] is \[\[d, -b]\[-c, a]] / (ad - bc)\]. Thus, \(A^{-1}\) will be \[\[1, -1]\[-3, 2]] / (2 - 3)\] which results in \[\[-1, 1]\[3, -2]]\].

Step 2: Compute B Inverse

Similarly, we calculate the inverse of the matrix \(B\). Thus, \(B^{-1}\) will be \[\[2, -7]\[-1, 4]] / (8 - 7)\] which results in \[\[-2, 7]\[1, -4]]\].

Step 3: Compute the product of A and B matrices

Matrix multiplication of \(A\) and \(B\) can be done using the element-wise formula, where each element of the new matrix is the sum of the products of the corresponding row and column elements from \(A\) and \(B\). This results in a new matrix: \[\[6, 21]\[7, 29]]\].

Step 4: Compute the inverse of AB

The inverse of \(AB\) can then be computed using the previously described formula, resulting in a matrix \[\[29, -21]\[-7, 6]] / (174 - 147)\] which simplifies to \[\[-29, 21]\[7, -6]]\].

Step 5: Compute \(A^{-1} B^{-1}\) and \(B^{-1} A^{-1}\)

We then compute \(A^{-1} B^{-1}\) and \(B^{-1} A^{-1}\) which will yield two different matrices. \(A^{-1} B^{-1}\) equals \[\[-8, 27]\[2, -7]]\] whereas \(B^{-1} A^{-1}\) equals \[\[-8, 7]\[27, -22]]\]. The result shows that matrix multiplication is not commutative.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra. Unlike regular multiplication of numbers, you can't just multiply matrices element by element. Instead, each element in the resulting matrix is obtained by taking the dot product of rows and columns.
To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix. For example, to multiply matrix \(A\) of size 2x2 with matrix \(B\), which is also 2x2, you perform the following steps:

• You take each row of matrix \(A\) and multiply the elements of that row by the corresponding column elements of matrix \(B\).
• Then sum up the results to form a single element in the resulting matrix.

This way, when multiplying two 2x2 matrices, you will end up with another 2x2 matrix. This resulting matrix contains the sum of products from the combinations of rows and columns.

Commutative Property

The commutative property is highly recognized in arithmetic for operations like addition and multiplication of numbers. However, when it comes to matrices, the commutative property isn't applicable.
This means that for two matrices \(A\) and \(B\), generally, \(A \cdot B eq B \cdot A\). Essentially, you can't swap the matrices around when multiplying them and expect the outcome to be the same.
In the previous exercise, when calculating both \(A^{-1} B^{-1}\) and \(B^{-1} A^{-1}\), we found two different results:

• \(A^{-1} B^{-1} = \begin{bmatrix} -8 & 27 \ 2 & -7 \end{bmatrix}\)
• \(B^{-1} A^{-1} = \begin{bmatrix} -8 & 7 \ 27 & -22 \end{bmatrix}\)

This confirms that matrix multiplication is not commutative, a key point to remember when working with matrices.

2x2 Matrices

2x2 matrices are among the simplest forms of matrices, often used to illustrate basic concepts like matrix operations due to their manageable size and simplicity.
A 2x2 matrix looks like this:\[\begin{bmatrix} a & b \ c & d \end{bmatrix}\]

• It has 2 rows and 2 columns.
• Many properties of larger matrices can be observed in 2x2 matrices, making them great for learning and exploration.

2x2 matrices can be added, subtracted, and multiplied as long as they conform to the basic rules of matrix operations. In the context of finding inverses, 2x2 matrices provide a simpler framework to practice these calculations due to their small size.

Inverse of a Matrix

The inverse of a matrix, if it exists, is akin to a reciprocal of a number. For a matrix to have an inverse, it must be square (same number of rows as columns) and its determinant must not be zero.
The formula for finding the inverse of a 2x2 matrix \(\begin{bmatrix} a & b \ c & d \end{bmatrix}\) is:\[\frac{1}{ad - bc} \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\]

• The determinant \(ad-bc\) should not be zero, as division by zero is undefined.
• The process involves swapping the positions of \(a\) and \(d\), and changing the signs of \(b\) and \(c\).

Inverting a matrix is not always possible, but for those that are invertible, it allows you to reverse the effects of a matrix multiplication operation, much like dividing by a number. Understanding inverses is crucial for solving matrix equations.