Question

If \(A\) is \(3 \times 3\) and invertible, compute \(\operatorname{det}\left(-A^{2}(\operatorname{adj} A)^{-1}\right)\)

Short answer

The determinant is \(-1\).

Step-by-step solution

Understanding the Problem

We are given a square matrix \( A \) of size \( 3 \times 3 \), which is invertible. We are tasked with finding the determinant of the matrix expression \( -A^2 (\operatorname{adj} A)^{-1} \).

Recognizing Properties of Determinants

Recall that for any scalar multiple of a matrix \( c \, A \), the determinant can be computed as \( \operatorname{det}(c \, A) = c^n \, \operatorname{det}(A) \), where \( n \) is the size of the matrix. Since \( A \) is \( 3 \times 3 \), \( n=3 \). Further recall that \( \operatorname{adj}(A) = \operatorname{det}(A) \, A^{-1} \).

Express (\operatorname{adj} A)^{-1} in terms of A

Since \( \operatorname{adj}(A) = \operatorname{det}(A) \, A^{-1} \), we find that \( (\operatorname{adj} A)^{-1} = \operatorname{det}(A)^{-1} (A) \).

Substitute and Simplify the Expression

Substitute \( (\operatorname{adj} A)^{-1} = \operatorname{det}(A)^{-1} (A) \) into the expression for the matrix: \(-A^2 (\operatorname{adj} A)^{-1} = -A^2 \cdot \operatorname{det}(A)^{-1} \cdot A^{-1} \). This simplifies to \(- (1/\operatorname{det}(A)) A \).

Compute the Determinant of the Product

The determinant of a product of matrices is the product of their determinants. Thus,\[\operatorname{det}(- (1/\operatorname{det}(A)) A) = - \frac{1}{\operatorname{det}(A)} \cdot \operatorname{det}(A) = -1.\]

Final Result

The determinant of the matrix \(-A^2 (\operatorname{adj} A)^{-1}\) is \(-1\).

Key concepts

Matrix adjugate

The matrix adjugate, often referred to as the adjugate or classical adjoint of a matrix, is an important concept in linear algebra. It is closely tied to the operations involving inverses and determinants. For a square matrix \( A \), the adjugate of \( A \), denoted as \( \operatorname{adj}(A) \), comprises the cofactors of the original matrix arranged in a transposed manner.
This means that each element of \( \operatorname{adj}(A) \) is the determinant of a minor of \( A \), with alternating signs, reflecting how cofactors are computed. The adjugate is especially useful because it relates to the inverse of a matrix through the formula:

• \( A^{-1} = \frac{\operatorname{adj}(A)}{\operatorname{det}(A)} \)

Understanding this relationship helps in scenarios where calculating the inverse directly seems hard, as the adjugate can simplify things by leveraging the properties of determinants.

Matrix inverse

The matrix inverse is a fundamental concept, especially when dealing with systems of equations or analyzing transformations. The inverse of a matrix \( A \), denoted as \( A^{-1} \), is a matrix such that the product of \( A \) and \( A^{-1} \) yields the identity matrix. Mathematically, this can be expressed as:

• \( A A^{-1} = I \)
• \( A^{-1} A = I \)

An important condition for the existence of an inverse is that the matrix must be invertible, meaning its determinant is non-zero. If the determinant is zero, the matrix is singular and doesn't have an inverse. Moreover, the inverse can be computed using the adjugate as mentioned before. This property is widely used in solving matrix equations, where rearranging and simplifying expressions can significantly reduce complexity.

Determinants properties

The determinant is a scalar value that provides insights into various aspects of a matrix, such as invertibility, volume transformation, and system of equations solutions. Key properties of determinants that play significant roles include:

• For a given matrix \( A \), if a scalar \( c \) multiplies a matrix, then \( \operatorname{det}(cA) = c^n \operatorname{det}(A) \) where \( n \) is the order of the matrix.
• The determinant of an invertible matrix is non-zero.
• The determinant of a product of matrices is the product of their determinants: \( \operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B) \).

These properties make determinants powerful tools in matrix algebra, influencing concepts like eigenvalues, transformations, and more. Students often rely on these properties to simplify complex matrix expressions and unravel intricate mathematical scenarios.

Matrix algebra

Matrix algebra is a cornerstone of linear algebra, offering computational tools and frameworks to solve system equations, perform transformations, and more. Different operations, such as addition, multiplication, inversion, and determinants, form the foundation of this field. Here are a few key concepts involved:

• Matrix addition and subtraction: Performed element-wise when matrices have the same dimensions.
• Matrix multiplication: Involves dot products of rows and columns, adhering to dimension compatibility for multiplication.
• Transpose of a matrix: Resulting in rows becoming columns and vice versa, useful in symmetric properties.

Understanding these operations can help solve complex problems, optimize calculations, and provide clarity in mathematical models. Matrix algebra's utility spans across computer graphics, physics, engineering, and more, making it an essential component of modern mathematics.