Problem 26 - Precalculus

Question

Let \(A\) be an arbitrary invertible \(3 \times 3\) matrix. (a) Show that \(\operatorname{det}\left(I_{3}\right)=1\). (See footnote \(^{8}\) below.) (b) Using the facts that \(A A^{-1}=I_{3}\) and \(\operatorname{det}\left(A A^{-1}\right)=\operatorname{det}(A) \operatorname{det}\left(A^{-1}\right),\) show that $$\operatorname{det}\left(A^{-1}\right)=\frac{1}{\operatorname{det}(A)}$$

Step-by-Step Solution

Answer

(a) \( \operatorname{det}(I_{3}) = 1 \). (b) \( \operatorname{det}(A^{-1}) = \frac{1}{\operatorname{det}(A)} \).

Step 1: Understand the Identity Matrix

The identity matrix for a 3x3 matrix, denoted as \( I_{3} \), is a matrix with 1s on the diagonal and 0s elsewhere. It looks like this: \[ I_{3} = \begin{pmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{pmatrix}. \]

Step 2: Calculate the Determinant of the Identity Matrix

To find \( \operatorname{det}(I_{3}) \), we use the formula for the determinant of a 3x3 matrix. The determinant of a diagonal matrix like \( I_{3} \) is the product of its diagonal elements: \( 1 \times 1 \times 1 = 1 \). Thus, \( \operatorname{det}(I_{3}) = 1 \).

Step 3: Recognize the Relation Between Matrix and Its Inverse

For any invertible matrix \( A \), the product of \( A \) and its inverse results in the identity matrix: \( AA^{-1} = I_{3} \). We also know that \( \operatorname{det}(AA^{-1}) = \operatorname{det}(I_{3}) = 1 \).

Step 4: Use the Property of Determinants for Products

The determinant of a product of matrices is the product of their determinants. Thus, \( \operatorname{det}(AA^{-1}) = \operatorname{det}(A) \times \operatorname{det}(A^{-1}) \). Since we know \( \operatorname{det}(AA^{-1}) = 1 \), we equate \( 1 = \operatorname{det}(A) \times \operatorname{det}(A^{-1}) \).

Step 5: Solve for \( \operatorname{det}(A^{-1}) \)

Rearranging the equation \( 1 = \operatorname{det}(A) \times \operatorname{det}(A^{-1}) \) gives \( \operatorname{det}(A^{-1}) = \frac{1}{\operatorname{det}(A)} \). This shows that the determinant of the inverse is the reciprocal of the determinant of \( A \).

Key Concepts

Invertible Matrix

In linear algebra, an invertible matrix is a square matrix that has an inverse. In other words, an invertible matrix is one that can be multiplied by its inverse to yield the identity matrix. This is an important property because if a matrix is invertible, you can use its inverse to solve linear equations and perform other operations.

For a matrix to have an inverse, it must be square, meaning it has the same number of rows and columns, and it must have a non-zero determinant. If a matrix has a determinant equal to zero, then it is not invertible, also known as singular. Essentially, the determinant provides crucial information about a matrix's invertibility. Understanding invertible matrices is essential, as they play a fundamental role in various applications across mathematics and engineering.

Identity Matrix

The identity matrix is a special type of square matrix that serves as the multiplicative identity in matrix algebra. It is denoted by the letter "I" or specifically, as in this case, by "I₃" for a 3x3 matrix. An identity matrix has ones on its main diagonal and zeros elsewhere, looking like this for a 3x3 configuration:

\[I_{3} = \begin{pmatrix} 1 & 0 & 0 \0 & 1 & 0 \0 & 0 & 1 \end{pmatrix}.\]

When any matrix is multiplied by the identity matrix, it remains unchanged. This is similar to how multiplying any number by one results in the original number. In the context of matrix algebra, the identity matrix plays a vital role in defining invertible matrices, as any matrix multiplied by its inverse results in the identity matrix.

Matrix Inverse

The matrix inverse is the counterpart to a matrix in multiplication, denoted often as \(A^{-1}\). When you multiply a matrix \(A\) by its inverse, you end up with the identity matrix \(I\), i.e., \(AA^{-1} = I\). Calculating the inverse involves several steps and depends on the matrix's determinant.

When \(A\) is invertible, its inverse can be used to solve systems of linear equations, among other operations. For instance, if you know \(Ax = b\), where \(x\) is a vector of variables, you can find \(x\) by multiplying both sides by \(A^{-1}\) to get \(x = A^{-1}b\). This is a simple yet powerful application of the inverse matrix, making it indispensable in many domains, such as computer graphics, engineering, and statistics.

Properties of Determinants

Determinants are a scalar value related to a matrix, providing essential insights into the matrix's properties and behavior. They play a significant role in determining whether a matrix is invertible. Some important properties of determinants include:

• Multiplicative Property: The determinant of a product of matrices is the product of their determinants. For matrices \(A\) and \(B\), \(\operatorname{det}(AB) = \operatorname{det}(A)\operatorname{det}(B)\).
• Determinant of the Identity Matrix: The determinant of an identity matrix is always 1, regardless of its size.
• Inverse Relationship: For an invertible matrix \(A\), the determinant of the inverse matrix \(A^{-1}\) is the reciprocal of the determinant of \(A\), \(\operatorname{det}(A^{-1}) = \frac{1}{\operatorname{det}(A)}\).
• Determinant Zero: If the determinant of a matrix is zero, the matrix is not invertible.

The properties of determinants are essential building blocks in matrix theory, helping us to solve systems of equations and analyze geometric transformations.