Problem 89 - College Algebra

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Give an example of a \(2 \times 2\) matrix that is its own inverse.

Step-by-Step Solution

Answer

Yes, there are examples of \(2 \times 2\) matrices that are their own inverses. For instance, the identity matrix \(\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\) is one. Another is \(\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}\).

Step 1: Understanding The Problem

Understand what an inverse matrix is and how a matrix can be its own inverse. This would mean that for a matrix \(A\), when it gets multiplied by itself (\(A^2\)), the result is the identity matrix.

Step 2: Try with an example

Now suppose we have such a \(2 \times 2\) matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\). If \(A\) is indeed its own inverse, then multiplying it by itself should yield an identity matrix \(\begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\) . This gives us the equations \(a^2+bc = 1\), \(ab + bd = 0\), \(ac + dc = 0\), and \(bc + d^2 = 1\). Using these equations, we can find the values for a, b, c, d that satisfy all of them.

Step 3: Find Suitable Values

Solving the equations from Step 2, one example of a solution would be \(a=d=1\), and \(b=c=0\). This provides us the matrix \(A = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}\), which is indeed the identity matrix, and so is its own inverse. Also, another example could be a=d=0 and b/c=-1. Then a matrix \(A = \begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}\) is also its own inverse.

Key Concepts

Identity Matrix

An identity matrix is a special type of square matrix that plays a similar role to the number 1 in regular arithmetic. It is denoted by the letter "I". In a 2x2 identity matrix, all the entries along the diagonal from the top left to the bottom right are 1, while all other entries are 0.

The 2x2 identity matrix looks like this:

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

This matrix is important because when any matrix is multiplied by an identity matrix, it remains unchanged, much like how multiplying any number by 1 leaves the number unchanged.

In terms of inverse matrices, if a matrix multiplied by itself results in the identity matrix, then that matrix is said to be its own inverse.

Matrix Multiplication

Matrix multiplication is a process to multiply two matrices, resulting in another matrix. For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second matrix.

In the case of 2x2 matrices, if we have matrices \(A\) and \(B\) such that:

• \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\)
• \(B = \begin{pmatrix} e & f \ g & h \end{pmatrix}\)

The product \(AB\) is computed as:

• First row: \( (ae + bg, af + bh) \)
• Second row: \( (ce + dg, cf + dh) \)

Matrix multiplication is not commutative; that is, \(AB\) is not necessarily equal to \(BA\). Understanding this concept is crucial in topics like solving matrix equations and finding inverse matrices.

2x2 Matrix

A 2x2 matrix is a simple type of matrix with two rows and two columns. Such matrices are crucial in linear algebra and applied mathematics because they're easy to understand, yet form the foundation for more complicated operations.

In a 2x2 matrix \(A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\), the elements \(a, b, c,\) and \(d\) can represent coefficients in systems of equations, entries of transformation matrices, or other mathematical structures.

These matrices can be easily added, subtracted, and multiplied using well-defined operations. Additionally, the determinant of a 2x2 matrix, given by \(ad - bc\), can be used to determine if the matrix is invertible. If the determinant is zero, the matrix does not have an inverse.

Matrix Equation

A matrix equation is an equation in which the variables are matrices. Similar to solving regular algebraic equations, the goal is often to find a specific matrix that satisfies the equation. They may look complicated but often follow straightforward principles.

Consider the equation \(AX = I\), where \(A\) is a given matrix and \(I\) is the identity matrix. Solving this equation involves finding the matrix \(X\), which, when multiplied by \(A\), results in the identity matrix. This involves computing the inverse of \(A\) so that \(X\) becomes \(A^{-1}\).

Solving matrix equations often involves finding the inverse of a matrix. For 2x2 matrices, if the entries satisfy certain equations, they can also be their own inverse, as shown with matrices like \(\begin{pmatrix} 0 & 1 \ -1 & 0 \end{pmatrix}\). This is a unique property worth verifying through operations and solutions.