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Chapter 8: Problem 54

If \(P\) is non-singular matrix, then value of \(\operatorname{adj}\left(P^{-1}\right)\) in terms of \(P\) is a. \(P A P \mid\) b. \(P|P|\) c. \(P\) d. none of these

Short Answer

Expert verified
The value of \( \operatorname{adj}(P^{-1}) \) is \( P \).

Step by step solution

01

Understanding the Problem

We are given that \( P \) is a non-singular matrix and we need to find the value of \( \operatorname{adj}(P^{-1}) \) in terms of \( P \). This is a problem related to properties of matrices, specifically inverse and adjugate matrices.
02

Review Matrix Properties

Recall that \( P \) is non-singular means \( P \) is invertible, i.e., \( \det(P) eq 0 \). One important property is that for any invertible matrix \( A \), \( \operatorname{adj}(A) = (\det A)A^{-1} \). We will use this property along with the fact that \( P \) is non-singular.
03

Expressing Adjugate in Inverse Terms

The adjugate of a matrix's inverse is related to the original matrix by: \( \operatorname{adj}(A^{-1}) = (\operatorname{adj}(A))^{-1} \cdot (\det A)^{-1} \) for any invertible matrix \( A \). Applying this to \( P \), we find \( \operatorname{adj}(P^{-1}) = (\operatorname{adj}(P))^{-1} \cdot (\det P)^{-1} \).
04

Substitute Known Identities

We know that \( \operatorname{adj}(P) = (\det P) P^{-1} \). Therefore, \( \operatorname{adj}(P)^{-1} = \det P^{-1} P \). So substituting, we get: \( \operatorname{adj}(P^{-1}) = \det P^{-1} P \cdot \det P^{-1} = P \).
05

Final Simplification

After simplifying the expression, we are left with the value of \( \operatorname{adj}(P^{-1}) = P \). Thus, the correct option is \( P \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Matrix Inverse
The concept of a matrix inverse is akin to finding the reciprocal of a function or a number. If you have a matrix \( P \), its inverse (denoted as \( P^{-1} \)) is the matrix that, when multiplied with \( P \), yields the identity matrix. Think of the identity matrix as the number 1 in linear algebra, meaning that any matrix multiplied by its inverse gets us back to the starting point, much like multiplying a number by its reciprocal gives you one.
  • The formula for a matrix inverse is: \( P \cdot P^{-1} = I \), where \( I \) is the identity matrix.
  • Not every matrix has an inverse. To have an inverse, the matrix must be square (same number of rows and columns) and non-singular (we鈥檒l explain what this means in a later section).
  • The inverse can be calculated through various methods such as row operations, using the adjugate method, or specialized algorithms.
Having mastery over finding and understanding matrix inverses is crucial in linear algebra as they are foundational in solving sets of linear equations and transformations.
Matrix Determinant
The determinant of a matrix, often denoted as \( \det(P) \), is a special number that can be calculated from its elements. Think of the determinant like an indicator of some essential properties of the matrix.
  • If the determinant is zero, the matrix is termed as 'singular', meaning it does not have an inverse.
  • For a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \ c & d \end{pmatrix} \), the determinant is given by \( ad - bc \).
  • Determinants also have geometric interpretations, like scaling factors for volumes when matrices represent linear transformations.
In essence, determinants are used in various computations in linear algebra, from determining the solvability of systems of equations to aiding in the computation of other matrix properties.
Non-Singular Matrix
A non-singular matrix is a matrix that can be inverted; it is also referred to as an invertible or non-degenerate matrix. In simpler terms, if a matrix is non-singular, it means you can "reverse" or "undo" the transformation it represents.
  • A matrix is non-singular if its determinant is non-zero. This is the key characteristic that separates singular matrices from non-singular ones.
  • Non-singular matrices are important in computational mathematics and engineering because they ensure that solutions to matrix equations exist and are unique.
  • They appear frequently in real-world problems, such as those involving equilibrium states in physical and economic systems.
Understanding whether a matrix is non-singular informs us about the properties of the system of equations or transformations it may represent.
Linear Algebra
Linear algebra is a branch of mathematics focusing on vectors, vector spaces, linear mappings, and systems of linear equations. It provides the theoretical underpinning for many scientific computations involving these concepts.
  • Central to linear algebra are matrices and operations such as addition, multiplication, and finding inverses.
  • The concepts of eigenvalues and eigenvectors, which arise from linear transformations, are key studies within linear algebra.
  • Most real-world problems, particularly those involving optimization, data science, machine learning, and physics, employ linear algebra techniques for problem-solving.
By listening to the orchestra of matrices, determinants, and inverses, linear algebra helps in understanding and navigating multi-dimensional spaces elegantly and effectively. It builds a language for describing and solving problems that might seem complex at first glance.

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Most popular questions from this chapter

If \(a, \beta, \gamma\) are three real numbers and \(A=\left[\begin{array}{ccc}1 & \cos (\alpha-\beta) & \cos (\alpha-\gamma) \\ \cos (\beta-\alpha) & 1 & \cos (\beta-\gamma) \\ \cos (\gamma-\alpha) & \cos (\gamma-\beta) & 1\end{array}\right]\), then which of following is/are true? a. \(A\) is singular b. \(A\) is symmetric c. \(A\) is orthogonal d. \(A\) is not invertible

If \(A\) and \(B\) are two non-zero square matrices of the same order such that the product \(A B=O\), then a. both \(A\) and \(B\) must be singular b. exactly one of them must be singular c. both of them are non-singular d. none of these

If \(A=\left[\begin{array}{cc}0 & -\tan \alpha / 2 \\ \tan \alpha / 2 & 0\end{array}\right]\) and \(I\) is a \(2 \times 2\) unit matrix, then \((I-A)\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \sin \alpha\end{array}\right]\) is \(\begin{array}{ll}\text { a. }-I+A & \text { b. } I-A\end{array}\) c. \(-I-A\) d. none of these

If \(A\) and \(B\) are square matrices of order \(n\), then \(A-\lambda I\) and \(B-\lambda I\) commute for every scalar \(\lambda\), only if a. \(A B=B A\) b. \(A B+B A=O\) c. \(A=-B\) d. none of these

The number of diagonal matrix \(A\) of order \(n\) for which \(A^{3}=A\) is a. I b. 0 c. \(2^{n}\) d. 3 "
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