Question

Use the following matrices. $$A=\left[\begin{array}{rr}4 & 3 \\ -2 & 1\end{array}\right], B=\left[\begin{array}{rr}1 & -2 \\ 3 & 4\end{array}\right], C=\left[\begin{array}{rr}13 & 4 \\ 1 & 8\end{array}\right]$$ Find \(C^{-1}, A^{-1} \cdot B^{-1}\), and \(B^{-1} \cdot A^{-1} .\) What do you observe about the three resulting matrices?

Short answer

These steps result in the calculation, through inversion and multiplication, of the matrices \(C^{-1}\), \(A^{-1} \cdot B^{-1}\), and \(B^{-1} \cdot A^{-1}\). Remember to note any patterns or interesting relationships between the resulting matrices.

Step-by-step solution

Compute Inverse of Matrix C

The inverse of a 2x2 matrix \(\left[\begin{array}{cc}a & b \ c & d\end{array}\right]\) is given by \(\frac{1}{ad-bc}\left[\begin{array}{rr}d & -b \ -c & a\end{array}\right]\). Apply this formula to matrix C.

Compute Inverses of Matrices A and B

Use the same formula to compute the inverses of matrices A and B.

Compute the Product \(A^{-1} \cdot B^{-1}\)

Now compute the product of the inverses of matrices A and B. Note that matrix multiplication is not commutative, which means that A times B is not necessarily the same as B times A.

Compute the Product \(B^{-1} \cdot A^{-1}\)

Compute the product of \(B^{-1} \cdot A^{-1}\) and compare the results of steps 3 and 4.

Observe the Patterns about the Results

Analyze the results of \(A^{-1} \cdot B^{-1}\) and \(B^{-1} \cdot A^{-1}\) and compare them with \(C^{-1}\), to see any possible patterns.

Key concepts

2x2 Matrix

A 2x2 matrix is a simple and quite common type of matrix that has two rows and two columns. Matrices are often used in various calculations in mathematics, physics, and computer science because they provide a concise way to represent and manipulate data. In a 2x2 matrix, which can be represented as follows:\[\begin{bmatrix}a & b \c & d\end{bmatrix}\]You have four elements arranged in a compact square format. Each element has its specific location in the matrix grid: \(a\), \(b\), \(c\), and \(d\) are placeholders for any numerical values or expressions. Understanding how to work with 2x2 matrices is vital because they often serve as building blocks for more complex matrix operations.2x2 matrices are straightforward yet powerful, allowing us to perform operations such as addition, subtraction, and multiplication. Moreover, they help in solving systems of equations and transforming geometric shapes, making them foundational in linear algebra and related fields.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra used to combine matrices. However, it's essential to note that this operation is not as straightforward as multiplying numbers. For matrices, the rule is that you can multiply them only if the number of columns in the first matrix equals the number of rows in the second matrix. For example, a 2x2 matrix can multiply another 2x2 matrix. Here's how it works:If we have two matrices, \[A = \begin{bmatrix} a & b \ c & d \end{bmatrix},B = \begin{bmatrix} e & f \ g & h \end{bmatrix},\]Their product AB is calculated as:\[AB = \begin{bmatrix} ae + bg & af + bh \ ce + dg & cf + dh \end{bmatrix}.\]This calculation involves a series of dot products between the rows of the first matrix and the columns of the second matrix. Due to this reason, the order in which you multiply matrices matters significantly. For instance, \(A \cdot B\) may not equal \(B \cdot A\). This property, non-commutative nature, is a key consideration when performing matrix operations, especially inversions and transformations in computational tasks or geometric mappings.

Inverse Matrices

The inverse of a matrix is akin to finding a reciprocal for a number. For a 2x2 matrix, the inverse allows you to undo the transformation such a matrix applies. However, not all matrices have inverses, and to find the inverse, a matrix must meet a critical condition: it must be non-singular, meaning its determinant should not be zero.To find the inverse of a 2x2 matrix:For \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix} \), its determinant \(det(A)\) is calculated as \(ad - bc\). The inverse, denoted by \(A^{-1}\), is found using the formula:\[A^{-1} = \frac{1}{ad-bc}\begin{bmatrix} d & -b \ -c & a \end{bmatrix}.\]This formula essentially changes the positions of \(a\) and \(d\), and negates the values \(b\) and \(c\). When you multiply a matrix by its inverse, you obtain the identity matrix, much like multiplying a number by its reciprocal results in 1.When working with inverse matrices, especially in equations, it's important to remember that matrix multiplication is not commutative:

• \(A^{-1} \cdot B^{-1}\) often gives different results than \(B^{-1} \cdot A^{-1}\).
• Thus, when reversing the order, you must carefully respect the order of multiplication.

Finding inverse matrices and performing such operations are key in solving linear systems and in transformations in vector space.